A REVIEW OF BOLTZMANN EQUATION WITH

Kinetic and Related Models c

American Institute of Mathematical Sciences Volume 2, Number 4, December 2009

doi:10.3934/krm.2009.2.xx pp. 1–XX

A REVIEW OF BOLTZMANN EQUATION WITH SINGULAR KERNELS

Radjesvarane Alexandre Irenav, French Naval Academy 29240 Brest Lanveoc-Poulmic, France

(Communicated by Tong Yang) Abstract. We review recent results about Boltzmann equation for singular or non cutoff cross-sections. Both spatially homogeneous and inhomogeneous Boltzmann equations are considered, and ideas related to Landau equation are explained. Various technical tools are presented, together with applications to existence and regularization issues.

Contents 1. Introduction 2. Spatially homogeneous Boltzmann equation: Introduction 2.1. Some previous results 2.2. Where are the difficulties? 3. Pseudo-differential formulations (ω−representation) 4. The cancellation Lemma 5. Coercivity results 6. Spatially homogeneous Boltzmann equation: Maxwellian case 6.1. A Littlewood-Paley proof 6.2. Further improvements 7. Functional estimates for the collision operator 7.1. Besov type estimates 7.2. A simple proof 8. Spatially homogeneous Boltzmann equation: Non Maxwellian case 8.1. A first conditional result 8.2. Further improvements 9. Spatially homogeneous Landau equation: Regularity 10. Back to coercivity: Linear or linearized case 10.1. The linear Boltzmann case (ω representation) 10.2. The linearized Boltzmann and Landau case 11. Spatially inhomogeneous Landau equation: Existence and regularity 11.1. The perturbative framework of Mouhot and Neumann 11.2. Regularization properties for Landau equation 11.3. Comments on perturbative solutions

2 6 7 8 11 15 16 19 20 24 24 24 26 33 34 38 39 43 43 47 51 51 53 54

2000 Mathematics Subject Classification. Primary: 42B25, 35Sxx; Secondary: 35Qxx. Key words and phrases. Boltzmann equations, cutoff hypothesis, existence, harmonic analysis, hypoellipticity, singular kernels.

1

2

RADJESVARANE ALEXANDRE

12. Spatially inhomogeneous Boltzmann equation: Introduction 13. Inhomogeneous Boltzmann equation: “Travelling” solutions 13.1. An existence result 13.2. Maximum type principles 14. Hypoelliptic properties and kinetic equations 14.1. A model problem 14.2. An uncertainty principle 14.3. Some applications of the uncertainty principle 15. Spatially inhomogeneous Boltzmann equation: Hypoellipticity 15.1. Technical tools 15.2. Regularizing effect 15.3. Existence and uniqueness of local solutions 16. Spatially homogeneous Boltzmann equation: Other results 16.1. Weighted Lp estimates 16.2. Other results: Stability, uniqueness 17. Conclusions: Some general remarks and open questions Acknowledgments REFERENCES

57 57 58 60 62 62 64 66 70 72 74 79 79 79 85 87 91 91

1. Introduction. Standard physical explanations around Boltzmann equation are exposed in the books of Cercignani [34, 33], Chapman and Cowling [38]. Therefore, this review will not contain any physical insights. The singular kernels mentioned in the title refer to collisional kernels which do not satisfy Grad’s cutoff assumption [74], see below for a precise definition. These kernels will be also referred as non cutoff kernels. Though the main topic of this review is about this non cutoff case, a good knowledge and mathematical ideas from the cutoff case are extremely important. Some classical references concerned with the cutoff case at the mathematical level are Arlotti and Bellomo [21], Cercignani, Illner and Pulvirenti [37], the works of DiPerna and Lions [57, 58, 91, 92, 93, 96], see also the old works of Ukai [123, 124]. In fact, for non singular kernels, there has been recent progress on using energy type methods, by Yang Guo, Tai-Ping Liu, Tong Yang and Shih-Hsien Yu, and these works are now extremely important also for singular kernels as well; precise references will be given later. One may also consult Ukai and Yang [125], presenting some of these recent results on Boltzmann equation in the cutoff case, and also the survey by SaintRaymond [117], related to hydrodynamical limits. However, this last topic will not be considered here. Together with entropy-entropy and hypocoercivity methods, see [55, 134, 135, 136, 137, 138] for much more details, these tools are at the basis of the present mathematical framework of Boltzmann equation. The singular case reviewed here was already considered in the recent and mathematically oriented review by Villani [134], written in 2003, but mainly from the renormalized point of view. However, the last years have seen some further progress on this topic: this is the aim of this review to present some of these works. Nevertheless, [134] also contains many physical insights together with a huge number of references on works related to Boltzmann equation, for which we refer the reader, so that we can limit the length of the present paper.

BOLTZMANN EQUATION

3

Boltzmann equation equation reads as follows ∂t f (t, x, v) + v.∇x f (t, x, v) = Q(f, f )(t, x, v),

(1)

where f = f (t, x, v) is the unknown function and typicall represents a (probability) density of particles at time t ≥ 0, located around position x ∈ Rn , with velocity close to v ∈ Rn . If the right hand side of (1) was set to zero, then this equation would describe the free behavior of particles. One then obtains a transport equation, which seems extremely simple, but has been the subject of a number of important works, related to velocity averaging results and dispersion effects, see [29, 32, 117], and for which certainly more is to be understood towards applications for Boltzmann equation. Since we are not considering free particles, at least because of collisions, under some physical assumptions, the right hand side, called Boltzmann collision operator, is an operator acting only w.r.t. velocity variable v, and reads, forgetting other variables Z Z Q(f, f )(v) = dv∗ dσB(v − v∗ , σ)(f ′ f∗′ − f f∗ ). (2) Rn

S n−1

n−1

Here, S denotes the unit sphere of Rn , n ≥ 2 . B(v − v∗ , σ) is a given positive function, depending only on the type of interactions between particles; it is called v−v∗ the collisional cross section, and actually only depends on |v − v∗ | and on |v−v · σ, ∗| so we shall write v − v∗ v − v∗ · σ) and cos θ = · σ. B(v − v∗ , σ) = B(|v − v∗ |, |v − v∗ | |v − v∗ |

f , f∗ , f∗′ and f ′ are usual and shorthand notations for f = f (v), f∗ = f (v∗ ), f ′ = f (v ′ ) and f∗′ = f (v∗′ ). Finally, for a given couple of velocities (v, v∗ ) (pre or post velocities), the couple (v ′ , v∗′ ) is given by v + v∗ |v − v∗ | v + v∗ |v − v∗ | + σ, v∗′ = − σ, 2 2 2 2 so that we have the following (momentum and energy) conservation rules v′ =

v ′ + v∗′ = v + v∗ and |v ′ |2 + |v∗′ |2 = |v|2 + |v∗ |2 .

(3)

(4)

The important cases are n = 2 or n = 3, but most of the underlying calculations do not depend on the dimension. The mathematical theory of Boltzmann equation is mainly separated into two parts: 1) one can look to the spatially homogeneous equation, that is for solutions independent of position variable x; 2) or to the spatially inhomogeneous case, that is the full Boltzmann equation above. But there is also another mathematical (and physical) separation, depending on whether or not, for a fixed z ∈ Rn − {0}, the function σ 7→ B(|z|, σ) is integrable on the unit sphere S n−1 . At the noticeable exception of hard sphere collisions, this is never so, typically because of a high singularity for θ close to zero, corresponding to so called grazing collisions. This high singularity implies additional huge difficulties in the mathematical treatment of Boltzmann equation, and this explains why Harold Grad [74] has introduced a cutoff assumption for such collisional cross sections. Thus up to the

4


1990’s, the main assumption on usual works around Boltzmann equation was Grad’s cutoff assumption, namely that σ 7→ B(|z|, σ) was integrable on the unit sphere S n−1 , or any similar kind of integrability. Up to the 1990’s, the main works on singular kernels were extremely limited and mainly due to Arkeryd [20], Pao [113] and Ukai [121]. Let us note that in this last reference, existence of solutions is shown for the spatially inhomogeneous Boltzmann equation with singular kernels, in Gevrey class. A classical physical example of cross-sections is given by inverse power laws 1 interactions, namely for an interaction potential similar to φ(r) = s−1 , s > 2. r Here r denotes the intermolecular distance. In this case, though the cross section can only be computed implicitely, a good approximation is given by B(|v − v∗ |, cos θ) = |v − v∗ |γ b(cos θ), sinn−2 θb(cos θ) ∼θ→0 Kθ−1−ν

(5)

s − (2n − 1) s−5 2 , so that γ = if n = 3, and ν = if n = 3. s−1 s−1 s−1 In particular, it is never integrable on S 2 . While we are introducing standard vocabulary, at least when n = 3, cases γ > 0, γ = 0 and γ < 0 are resp. called hard, maxwellian and soft cases. Therefore, when we are talking about singular (or non cutoff ) cross sections, we are just mentioning the fact that our main assumption is that the cross section will never be integrable over the unit sphere, and typically are given by the above example (5). Up to that point, nothing special seemed to be attached with the singular case, but in the 1990’s, Desvillettes started showing that singular kernels lead to very different effects compared to the non singular case, see [42, 43, 44, 50]: solutions to Boltzmann spatially homogeneous equation enjoy regularization properties! Strongly related to this framework is another well known equation: Landau equation, taking into account all grazing collisions. It is important to notice that works around this latter equation can give (and in fact have given) better insights into Boltzmann equation. We should mention that there is still a lot of open questions around this last equation. We shall come back to Landau equation in later Sections, but we just mention here that the Landau equation is similar to Boltzmann equation but with a collision operator given by Z L Q (f, f )(v) = ∇v .( dv∗ a(v − v∗ )[f∗ (∇f ) − f (∇f )∗ ]dv∗ , (6) where γ =

Rn

where a(z) = |z|2 Φ(z)Πz⊥ , Πz⊥ being the orthogonal projection onto z ⊥ that is zi zj (Πz⊥ )i,j = δi,j − 2 . |z|

Function Φ which appears above is, in the simplest case, related to Boltzmann cross-section by the relation B(v − v∗ , cos θ) = Φ(v − v∗ )b(cos θ).

Turning to Boltzmann equation (1), as usual in the general theory of Partial Differential Equations, it is important to have (good) à priori estimates. The natural ones for Boltzmann equation (and up to now, the only known ones at the exception of one dimensional models) are based on symmetrization properties of the collision operator, and hold true for both singular or non singular kernels: v−v∗ using in particular the change of variables (v, v∗ , σ) 7→ (v ′ , v∗′ , k), with k = |v−v ∗|

BOLTZMANN EQUATION

5

which has unit Jacobian and is involutive, one finds that for suitable test functions φ = φ(v) Z    dvQ(f, f )(v)φ(v) = Z Z Z Rn (7) 1   − dv dv∗ dσ(f ′ f∗′ − f f∗ ){φ′ + φ′∗ − φ − φ∗ }. 4 Rn Rn S n−1

Also by integrating over variable x in the spatially inhomogeneous case, one finds the following conservation laws Z Z d f (t, x, v)φ(v)dvdx = 0 if φ(v) = 1 or vi , 1 ≤ i ≤ n, or |v|2 , (8) dt Rn Rn

which express the global conservation of mass, momentum and energy. Let us mention that for general initial data, and more precisely in the context of renormalised solutions introduced by Di Perna and Lions [58, 91] in the cutoff case, and by Alexandre and Villani in the non cutoff case [3, 16], it is still unknown if the corresponding solutions do satisfy momentum and energy local conservation laws. These local conservation laws read here, ∂t ρ + ∇x .(ρu) = 0, Z ∂t (ρu) + ∇x .( f v ⊗ vdv) = 0 Rn

and

Z ∂t (ρ|u| + n̺T ) + ∇x ( 2

Rn

f |v|2 dv) = 0.

A last important inequality, namely Boltzmann’s H-theorem, follows by choosing φ(v) = log f (v). Upon introducing the H-functional Z Z H(f ) = f log f (t, x, v)dvdx, (9) Rn Rn

and the entropy dissipation functional Z Z Z f ′ f∗′ 1 D(f ) = dv dv∗ dσB(v − v∗ , σ)(f ′ f∗′ − f f∗ ) log ≥ 0, 4 Rn f f∗ Rn S n−1 one finds also d H(f ) + dt

Z

D(f )dx = 0.

(10)

(11)

Rn

The representation of the collision operator above, given by (2), along with (3), is referred as the σ-representation. Another well known representation is given by the so-called ω-representation (which was used mostly before the 2000’s!) Z Z θ Q(f, f )(v) = dv∗ dω[f ′ f∗′ − f f∗ ]2n−2 sinn−2 ( )B(|v − v∗ |, cos θ), (12) 2 n n−1 R S where now we have the rules v ′ = v + (v∗ − v).ω ω and v∗′ = v∗ − (v∗ − v).ω ω.

(13)

Finally, a third and last representation, due initially to Carleman, is also useful: letting Ev,v−v′ be the hyperplane going through v and orthogonal to v − v ′ , using

6


the identity v − v∗ = 2v − v ′ − v∗′ , one has  Q(f, f )(v) =  Z Z  ′   dv v ′ − v∗′ ∗  ′ ′ dv ′ B(2v − v − v , )[.] ∗ ′ n−1 | v ′ − v∗′ | Rn Ev,v−v′| v − v |      with [.] ≡ [f (v∗′ )f (v ′ ) − f (v)f (v ′ + v∗′ − v)].

(14)

We shall mainly use the σ-representation, but at some places, the two other representations might be useful, and have proven to be important. For more explanation on these representations and their usefulness, see [134] and the references therein. This review is organized as follows. Section 2 will recall some previous and basic results for the Boltzmann spatially homogeneous equations, for singular kernels. In particular, we have included standard computations on the collision operator. Section 3 will show that this operator behaves as a fractional Laplacian operator. In Sections 4 and 5, we shall recall some of the results (cancellation and coercivity) from [9], which will play a crucial role in the rest of the review. We shall start to show regularization properties in Section 6, with the simplest case of the spatially homogeneous Boltzmann equation, but with Maxwellian type molecules. In order to extend these results to more general cross-sections, Section 7 will recall some functional estimates on Boltzmann collision operator which were obtained recently. These estimations are crucial for Section 8, where regularization properties are considered, still for the spatially homogeneous Boltzmann equation, but with more general cross-sections. Because of its close connection with Boltzmann equation, Landau spatially homogeneous equation will be considered in Section 9. Particularly, it should be noticed that results for Landau equation consider more physical cross-sections. Section 10 will review works related to Boltzmann linearized operator. For instance, these results play an important role in the study of perturbative solutions. As an illustration, Section 11 will deal with Landau spatially inhomogeneous equation, both for existence and regularization questions. We shall start reviewing the spatially inhomogeneous Boltzmann equation in the very short Section 12. Section 13 shows how to construct “travelling” type solutions, together with a partial and small regularity property. Because hypoellipticity questions are closely related to Boltzmann equation, we shall devote Section 14 to such problems and applications to model kinetic equations. Section 15 shows how to apply all the tools from the previous Sections to the complete Boltzmann equation. In Section 16, we shall get back to the spatially homogeneous Boltzmann equation and consider further types of estimations. Finally, we shall end by mentioning in Section 17 some general open questions and problems. In writing this review, we have decided to make several restricted choices in order to limit the length of the paper, and focus on special questions related to our own present thinking. We have tried to complement this point of view by mentioning a large number of recent works where one can find other important ideas. 2. Spatially homogeneous Boltzmann equation: Introduction. Let us first of all mention the works of Arkeryd [20], Goudon [72], Villani [130], see the reviews by Desvillettes [45, 46]. Further recent results, such as propagation of moments, unicity of solutions..., can be found in the works of Desvillettes, Mouhot [52, 51, 53]. We shall come back later on these works, together with regularization properties of solutions.

BOLTZMANN EQUATION

7

2.1. Some previous results. First of all, weak solutions can be defined by using the symmetrisation properties of the collision operators, in particular the duality formula (7). This leads to the usual notion of weak solutions, and eventually also using the entropy dissipation estimates, to the notion of H-solutions. That is, we are only using basic estimates (8) and eventually those implied by (10). One has then the following summary Theorem 2.1 (Desvillettes [45]). 1) If γ ∈]0, 1[ (hard potentials), then a weak solution exists. 2) If γ = 0 (maxwellian molecules) or γ ∈] − 2, 0[ (soft potentials), then a weak solution exists. 3) If γ ∈] − 3, −2[ (very soft potentials), then an H-solution exists. An important problem is to get estimates on moments with respect to velocity variable. As regards propagation of moments, it is known that if γ ∈]0, 1[ (hard potentials), then (polynomial) moments are propagated, remain bounded, and in fact there is immediate creation of arbitrary higher moments, for positive times. For other values of γ, it is known that moments are at least propagated, and bounded at least in the Maxwellian case. We state the following summary, about moments estimations (in fact there is a better result stated therein, for which we refer for more details). γ (cos θ), −3 ≤ γ ≤ 1 Theorem 2.2 (Villani [134]). Let B(v R π− v∗ , cos θ) = |v − v∗ | n−2 and the momentum control estimate 0 b(cos θ)(1 − cos θ) sin θdθ < +∞. Let f0 ∈ L12 (Rnv ) ≥ 0 be an initial datum with finite mass and entropy, and R f = f (t, v) a weak solution of Boltzmann equation, with decreasing kinetic energy f (t, v)|v|2 dv. Then this kinetic energy is automatically constant in time. Moreover, if we set Z Ms (t) = f (t, v)|v|s dv,

then 1. If γ = 0, then for all s > 2, ∀t > 0, Ms (t) < +∞ iff Ms (0) < +∞. If Ms (0) < +∞ then sup Ms (t) < +∞. t≥0

2. If γ > 0, then for all s > 2, ∀t0 > 0, sup Ms (t) < +∞. t≥t0

3. If γ < 0, then for any s > 2, ∀t < +∞, Ms (t) < +∞ iff Ms (0) < +∞. The above result applies to any weak solution, where we are using the duality formula mentioned previously, but for γ ≥ −2. For smaller values of γ, this duality formula is no more appropriate, and we need to shift to the notion of H-solutions. This notion was introduced by Villani [130], and is a way to encompass the kinetic singularity, i.e. bad exponent γ.

8


2.2. Where are the difficulties? Let us explain some of the difficulties related to the singular case, so that by the way, we shall justify this notion of (weak) solutions. Assume for a while that n = 3, and take our collision cross section as B(v − v∗ , σ) = |v − v∗ |γ b(cos θ), where R π 2 b(cos θ) behaves as above, for inverse power potentials, and thus satisfies θ b(cos θ)dθ < +∞ (recall that we are in dimension 3). 0 First of all, recall the different duality formulations of the collision operator, when tested against a suitable test function φ = φ(v). For instance, one possible choice is given by Z Z Z 1 dvQ(f, f )(v)φ(v) = dvdv∗ B(v − v∗ , σ)f f∗ {φ′ + φ′∗ − φ − φ∗ }. 2 R3 R3 R3 We note that, using Taylor’s formula, ′

′

′

′

φ − φ = (v − v).∇φ(v) + (v − v) ⊗ (v − v) : and

Z

φ′∗ − φ∗ = (v∗′ − v∗ ).∇φ(v∗ ) + (v∗′ − v∗ ) ⊗ (v∗′ − v∗ ) : Since v∗′ − v∗ = −(v ′ − v), it follows that

1 0

Z

(1 − t)D2 φ(v + t(v ′ − v))dt

0

1

(1 − t)D2 φ(v∗ + t(v∗′ − v∗ ))dt.

φ′ + φ′∗ − φ − φ∗ = O(|v − v∗ |2 θ ∧ 1).

This estimation is sufficient to kill small singularities 0 < ν < 1 and for γ ≥ −2, and therefore Q(f, f ) can be defined by duality in that case. What can we do for higher singularities, 1 ≤ ν < 2? After many preliminary works (precise references are given by Villani [130]), here is the answer. Without further difficulties, we may consider a more general framework, see for example [16]. Without loss of generality, we assume that B(v − v∗ , σ) is supported in the set (0 ≤ θ ≤ π/2), i.e. (v − v∗ , σ) ≥ 0. If not, we can reduce to this case replacing B by its symmetrized version B(v − v∗ , σ) = [B(v − v∗ , σ) + B(v − v∗ , −σ)]1(v−v∗ ,σ)>0 . as

We introduce the momentum transfer, associated with the collision cross section,  Z   M(|v − v∗ |) ≡ B(v − v∗ , σ)(1 − k · σ) dσ  S n−1 Z π2 (15)   B(|v − v∗ |, cos θ)(1 − cos θ) sinn−2 θ dθ.  = |S n−2 | 0

v − v∗ where the notations k = and cos θ = k · σ are used. |v − v∗ | 2 Since 1 − cos θ = 2 sin (θ/2) vanishes up to order 2 for θ close to 0, the above quantity is finite for the usual model case of cross sections. We introduce a regularity assumption on B w.r.t. the relative velocity variable z = v − v∗ by introducing, for z 6= 0, B ′ (z, σ) =

|B(λz, σ) − B(z, σ)| . (λ − 1)|z| 2

sup√

1 0 if |z| = 6 0. In particular, in the model case sinn−2 θ b(cos θ) ∼ Kθ−1−ν ,

B(v − v∗ , σ) = |v − v∗ |γ b(cos θ),

ν > 0, K > 0, Assumptions I, II and III (18), (20), (21) together allow the following range of parameters : γ ≥ −n,

0 ≤ ν < 2,

γ + ν < 2.

(22)

Now, we can get back to the bilinear Boltzmann operator Q(f, f ), defined by duality as follows. Letting ϕ(v) be a smooth test-function in the velocity variable, we define Z Z Z ′ Q(f, f )φ(v)dv = dv dv∗ f f∗ B(v − v∗ , σ)(ϕ − ϕ) dσ . Rn

R2n

S N −1

Note that we do not use the full duality formula from the previous Section. The above formula suggests to study, for given v∗ , the linear operator Z T : ϕ 7−→ B(v − v∗ , σ)(ϕ′ − ϕ) dσ. (23) S n−1

We shall get back to much more precise functional properties of this operator, but for the moment, the following Proposition is sufficient for our immediate purpose.

10


Proposition 2.3. Under Assumptions (18), (20) and (21), for all ϕ ∈ W 2,∞ (Rnv ), 1 |v − v∗ | |T ϕ(v)| ≤ kϕkW 2,∞ |v − v∗ | 1 + M(|v − v∗ |). 2 2

Moreover, for all α ∈ [0, 2] and ϕ ∈ W 2,∞ (Rnv ),

|T ϕ(v)| ≤ 2kϕkW 2,∞ (1 + |v − v∗ |)α Mα (|v − v∗ |),

where Mα is defined by formula (19). In particular, if B satisfies the hard potential case (or even with γ ≥ −2), and if f satisfies the usual a priori entropic bounds, then, for all R > 0, Q(f, f ) defined by duality as above belongs to L∞ ([0, T ]; W −2,1 (BR (v)))), where BR (v) denotes {v ∈ Rn , |v| ≤ R}. The above framework gives, with some additional works (in particular using an approximation by non singular cross sections), existence of weak solutions, in the case γ > −2. The difficulty still occurs for lower values of γ, that is γ > −3 (note by the way that the same problem does not occur in the spatially inhomogeneous and renormalized framework!). The trick, introduced by Villani [130], consists in changing the definition of solutions, shifting to so-called H-solutions, and using the entropy dissipation functional. We shall see later on that in a very loose sense, we are using an hidden but better regularity on f . First of all, since the entropy dissipation rate is bounded, we find the following a priori estimate Z TZ Z Z p p dvdv∗ dσB(v − v∗ , σ( f ′ f∗′ − f f∗ )2 ≤ C. 0

Rn Rn S n−1

Our task is to give a sense to a weak form of the collision operator, allowing for values of γ , γ > −3. At this step, two ways are possible, but both start from the simple formula p p p p f ′ f∗′ − f f∗ = ( f ′ f∗′ − f f∗ )( f ′ f∗′ + f f∗ ) p p p p p = ( f ′ f∗′ − f f∗ )[ f ′ f∗′ − f f∗ + 2 f f∗ ] and thus

p p p p p f ′ f∗′ − f f∗ = ( f ′ f∗′ − f f∗ )2 + 2( f ′ f∗′ − f f∗ ) f f∗ .

Therefore, we can write Z p Z p p p p 1 Q(f, f )φ(v)dv = − ( f ′ f∗′ − f f∗ )2 (φ′ −φ)− ( f ′ f∗′ − f f∗ ) f f∗ (φ′ −φ) 2 and also Z Z p p 1 Q(f, f )φ(v)dv = − ( f ′ f∗′ − f f∗ )2 (φ′ + φ′∗ − φ − φ∗ ) 4 Z p p p 1 − ( f ′ f∗′ − f f∗ ) f f∗ (φ′ + φ′∗ − φ − φ∗ ). 2 By symmetry, we deduce Z Z p p p Q(f, f )φ(v)dv = − ( f ′ f∗′ − f f∗ ) f f∗ (φ′ − φ) Z

BOLTZMANN EQUATION

and

Z

11

Z p p p 1 ( f ′ f∗′ − f f∗ ) f f∗ (φ′ + φ′∗ − φ − φ∗ ). 2 The second formula is well defined for weak solutions: use the entropy dissipation estimate displayed above, combined with Cauchy Schwartz inequality, and note that the square leads to a good cancellation. We refer to [130] for a complete study. Q(f, f )φ(v)dv = −

3. Pseudo-differential formulations (ω−representation). The previous Section has introduced some of the basic computations on the singular case for the spatially homogeneous Bolzmann equation. After these preliminaries, what was the state of the art in the 1990’s? We have recalled the (old) result of Ukai [122] for the spatially inhomogeneous Boltzmann equation, and mentioned previously the existence result of Villani [130] in the spatially homogeneous case. In the 1990’s, in a series of works [42, 43, 44], see also Proutière [115], Laurent Desvillettes showed that regularizing effects occur for solutions, and this is definitively different from the cutoff case, for which there is no such effect, see for example [30, 91, 92, 93]. On the other hand, following works from Lions and Villani [94, 128] on the spatially inhomogeneous Landau equation, strong compactness of solutions are true for weak solutions of this equation. Since at that time, it was known that Landau equation occurs as a grazing limit of Boltzmann equation, see [22, 40, 41, 72], and that Landau operator is a divergence type operator with a positive symmetric matrix, the question therefore arises if something similar is true for Boltzmann operator. This is the case, but through a pseudo-differential formulation of Boltzmann collision operator, as we shall now explain. Such a formulation has an obvious advantage: it shows very clearly that Boltzmann’s operator behaves as a fractional Laplacian, with the good sign, and thus explains what kind of effects one could hope for. However, the pseudo-differential approach is extremely intricate and hard to use, see for more details in [2, 5]. We start from the full bilinear operator Z

Z

˜ v − v1 |, | ( v − v1 , ω) |). dv1 dω{f (v ′ )g(v1′ ) − f (v)g(v1 )}B(| | v − v1 | (24) For convenience, we have assumed that velocity space dimension is 3 and we are also using the ω representation, see Section 1, with the following exact expression ˜ for the collisional kernel B Q(f, g)(v) ≡

R3v1

2 Sω

1 ˜ v − v1 |, | ( v − v1 , ω) |) ≡| v − v1 |γ B(| , v−v1 | v − v1 | | ( |v−v , ω) |ν 1|

(25)

and the critical exponents are defined by

s−5 s+1 , ν = ν(s) = , (26) s−1 s−1 where we shall always assume s > 2. More general cases can be dealt with but with not so explicit computations. Actually, the assumption (25) helps clarify the computations. First of all, one has [5] γ = γ(s) =

12


Lemma 3.1. Assume (25) and (26). Then the non linear Boltzmann operator Q writes as Z

Q(f, g)(v) =

R3h

2dh | h |ν+2

Z

E0,h

{f (v − h)g(α + v) − f (v)g(α + v − h)}×

×{| α |2 + | h |2 }

γ+ν 2

dα,

where E0,h denotes the plane containing 0 and orthogonal to h ∈ R3 . The proof of the above Lemma follows by using Carlemann representation, which was crucial for example in the study of the gain term in the cutoff case [91, 103, 140], which transforms the integral over S 2 into an integral over a plane. As a byproduct of the proof, we have also Z Z dv1 dω Q(f, g)(v) ≡ {f (v ′ )g(v1′ ) − f (v)g(v1 )} | v − v1 |γ+ν . (27) | v ′ − v |ν 2 R3v Sω 1

2(s − 2) > 0 as s > 2. s−1 ′ From v1 − v = (v1 − v ) + (v ′ − v) and (v1 − v ′ ).(v ′ − v) = 0, one may write

Note that γ + ν =

Q(f, g)(v) = Q1 (f, g)(v) + Q2 (f, g)(v),

(28)

where Q1 (f, g)(v) ≡

Z

R3v1

Z

2 Sω

{f (v ′ )g(v1′ ) − f (v)g(v1 )} | v1 − v ′ |γ+ν

dv1 dω , | v ′ − v |ν

(29)

and   

Q2 (f, g)(v) ≡

R

R3v1

R

2 Sω

{f (v ′ )g(v1′ ) − f (v)g(v1 )}×

  ×[{| v − v ′ |2 + | v ′ − v |2 } γ+ν dv1 dω 2 − | v − v ′ |γ+ν ] 1 1 |v ′ −v|ν .

(30)

Using the arguments involved in the proof of Lemma 3.1 (see [5]) and an obvious splitting, it follows that Q1 (f, g) = Q1,1 (f, g) + Q1,2 (f, g), where Q1,1 (f, g)(v) =

Z

R3h

2dh | h |ν+2

Z

E0,h

{f (v − h) − f (v)}g(α + v) | α |γ+ν dα

(31) (32)

and Q1,2 (f, g)(v) = f (v)

Z

2dh | h |ν+2

Z

E0,h

{g(α + v) − g(α + v − h)} | α |γ+ν dα. (33)

βg (u, v) ≡

Z

E0,u

g(α + v) | α |γ+ν dα.

R3h

For u ∈ S 2 , v ∈ R3 , let

Then clearly

BOLTZMANN EQUATION

Q1,1 (f, g)(v) =

Z

Q1,2 (f, g)(v) = f (v)

13

2dh h {f (v − h) − f (v)}βg ( , v), ν+2 |h| |h|

R3h

Z

R3h

2dh h h {βg ( , v) − βg ( , v − h)}, | h |ν+2 |h| |h|

(34) (35)

Q1 (f, g) = Q1,1 (f, g) + Q1,2 (f, g). An “explicit” form for Q1,1 is stated by [5]

Lemma 3.2. Let Q1,1 be given by (34). Then, there exists a fixed constant Cs′ depending only on s (or ν), such that (with the notations of pdo theory) Z ′ Q1,1 (f, g)(v) = −Cs dαg(α + v) | α |γ+ν | S(α).D |ν−1 (f )(v), R3α

where S(α) denotes the projection operator over the hyperplane E0,α . The value of Cs′ may be found below. Proof. Since βg (−u, v) = βg (u, v), making the change of variables h → −h, we find that Q1,1 (f, g)(v) =

Z

R3h

dh h {f (v − h) + f (v + h) − 2f (v)}βg ( , v). ν+2 |h| |h|

Shifting to polar coordinates, h = rω, with r =| h |, ω = Q1,1 (f, g)(v) =

Z

+∞

dr

Z

h , we get also |h|

1 {f (v + rω) + f (v − rω) − 2f (v)}βg (ω, v). (37) rν

dω

2 Sω

0

(36)

Next, letting fˆ(ξ) for the Fourier transform of f , (37) writes as Q1,1 (f, g)(v) =

Z

dξ fˆ(ξ)eiξ.v

R3ξ

Z

dωβg (ω, v){

2 Sω

Z

0

+∞

dr irξ.ω −irξ.ω (e +e −2)}. (38) rν

From classical homogeneity arguments, the curly brackets term above may be written as Z ∞ dr irξ.ω (e + e−irξ.ω − 2) ≡ −Cs | ξ.ω |ν−1 , (39) ν r 0 Z ∞ dr Cs ≡ (2 − eir − e−ir ). (40) rν 0 Getting back to (38), we obtain Z Z Q1,1 (f, g)(v) = −Cs dξ fˆ(ξ)eiξ.v dωβg (ω, v) | ξ.ω |ν−1 . (41) R3ξ

2 Sω

We notice also that

βg (ω, v) =

Z

R3α

dα δα.ω=0 g(α + v) | α |γ+ν ,

according to which (41) transforms into

(42)

14


Q1,1 (f, g)(v) = −Cs that is also (

Z

dξ fˆ(ξ)eiξ.v

R3ξ

Z

dω

2 Sω

R

Q1,1 (f, g)(v) = −

R3ξ

{Cs

and again, we define the Z {Cs

R

dξ

R

R3α

R3α

dαδα.ω=0 g(α + v) | α |γ+ν | ξ.ω |ν−1 ,

dαfˆ(ξ)eiξ.v g(α + v) | α |γ+ν ×

2 ,ω.α=0 Sω constant Cs′ as

2 ,ω.α=0 Sω

Z

dω | ξ.ω |ν−1 },

dω | ξ.ω |ν−1 } = Cs′ | S(α).ξ |ν−1 .

(43)

(44)

Turning to (43), we find

Q1,1 (f, g)(v) = −Cs′

Z

R3ξ

dξ

Z

R3α

dα | α |γ+ν g(α + v) | S(α).ξ |ν−1 fˆ(ξ)eiξ.v

(45)

which is nothing else than the expression given in the statement of the Lemma. The above arguments also apply to the analysis of Q1,2 Lemma 3.3. Let Q1,2 be the operator given by (35). Then, with the same constant Cs′ as above, one has Z Q1,2 (f, g)(v) = Cs′ .f (v). dα | α |γ+ν | S(α).D |ν−1 (g)(α + v). R3α

What have we shown? First of all, Q1,1 is some type of “negative” pseudodifferential operator with strictly positive order, that is too say, formally elliptic. We shall see in the next Section, that the analysis of Q1,2 is in fact given by the Cancellation Lemma, and is of order 0, so that again formally, does not contribute to an expected regularity. There remains to study Q2 , which is in fact a Boltzmann operator but with a cutoff type kernel. For this purpose, set ( v−v1 1 B c (| v − v1 |, | ( |v−v , ω) |) ≡ |v′ −v| ν 1| (46) γ+ν ′ 2 ′ 2 2 ×[{| v1 − v | + | v − v | } − | v1 − v ′ |γ+ν ]. Note that {| v1 − v ′ |2 + | v ′ − v |2 }

γ+ν 2

=| v − v1 |γ+ν and

v − v1 , ω) |2 }. | v − v1 | γ+ν s−2 γ+ν Of course B c ≥ 0, and since = ,0< < 1, we deduce that 2 s−1 2 1 1 c | B c (., .) | ≤ Bmax (., .) ≡ C , (47) | v1 − v ′ |2−(γ+ν) | v ′ − v |ν−2 which is also 1 1 c Bmax (., .) = C | v1 − v |γ . (48) γ+ν . v−v1 v−v1 1− 2 | ( |v−v1 | , ω) |ν−2 2 {1− | ( , ω) | } | v1 − v ′ |2 =| v1 − v |2 {1− | (

|v−v1 |

BOLTZMANN EQUATION

15

γ +ν c ) < 1, −1 < ν − 2 < 1 and −3 < γ < 1, Bmax and 2 c 2 therefore B are integrable over S , for fixed v1 − v. Recalling that 0 < 1 − (

4. The cancellation Lemma. We are now back to the σ−representation. Recalling the splitting of the previous Section 3, we now write more generally Z Z Z Z Q(f, f ) = dv∗ dσB(v−v∗ , σ)f∗′ [f ′ −f ]+f dv∗ dσB(v−v∗ , σ)(f∗′ −f∗ ), Rn S n−1

and we are interested in the map f 7→

Rn S n−1

Z Z

dv∗ dσB(v − v∗ , σ)(f∗′ − f∗ ).

Rn S n−1

This is the aim of the cancellation Lemma taken from Alexandre, Desvillettes, Villani and Wennberg [9], that we shall use in the following Sections. Roughly speaking, it says that this part of the Boltzmann operator makes perfectly good sense, without any further regularity assumptions on functions, though at first sight, one would ask for functions having at least a smoothness of order 1 or 2. Lemma 4.1 (Cancellation Lemma [9]). Let B be a nonnegative measurable kernel and Assumptions (18), (20) and (21) hold true. Then, for a.a. v ∈ Rn , Z Sf ≡ dv∗ dσ B(v − v∗ , σ)(f∗′ − f∗ ) = f ∗v S, (49) Rn ×S n−1

where

S(|z|) = |S

n−2

|

Z

π 2

n−2

dθ sin

0

1 θ B n cos (θ/2)

|z| , cos θ − B(|z|, cos θ). . cos(θ/2) (50)

In particular, S is a locally integrable function, n−4

2 2 |S(|z|)| ≤ nM(|z|) + |z|M′ (|z|) . cos2 (π/8)

We only give the very first lines of the proof of Lemma 4.1, and in particular display an important change of variables. Since S is defined (and has to be defined as such) as a principal value operator, computations will be done assuming B to be integrable. We perform the change of variables v∗ → v∗′ , for each v, σ fixed, which is well-defined on (cos θ > 0), with Jacobian given by ′ dv∗ 1 (k ′ · σ)2 dv∗ = 2n (1 + k · σ) = 2n−1 , where

v − v∗′ , |v − v∗′ |

θ 1 k ′ · σ = cos > √ . 2 2 √ ′ ′ Let ψσ : v∗ 7−→ v∗ , defined on (k · σ) > 1/ 2, and note that |v∗ − ψσ (v ′ )| = ′ |v − v∗ |/(k ′ · σ), that is |v − v∗ | |v∗ − ψσ (v)| = . kR· σ Applying this change of variable to the part Bf∗′ in the left-hand side of (49), one gets Z Z ′ ′ dv∗ B v − ψσ (v∗′ ), σ f∗′ dσ dv∗ B(v − v∗ , σ) f∗ = √ dσ dv∗ ′ dv n−1 n ′ S ×R k ·σ≥1/ 2 ∗ k′ =

16


= Noticing that

Z

√

k′ ·σ≥1/ 2

dσ dv∗′

2N −1 B v − ψσ (v∗′ ), σ f∗′ . ′ 2 (k · σ)

B v − ψσ (v∗′ ), σ = B |v − ψσ (v∗′ )|, k · σ = B |v − ψσ (v∗′ )|, 2(k ′ · σ)2 − 1 ,

and changing the name v∗′ for v∗ , (49) holds with Z 2n−1 2 S(|v − v∗ |) = B |v − ψ (v)|, 2(k · σ) − 1 dσ ∗ σ (k · σ)2 k·σ≥ √12 Z − dσB(|v − v∗ |, k · σ). k·σ≥0

The first part is then Z

|v − v∗ | 2 , 2(k · σ) − 1 k·σ k·σ≥ √12 Z π4 2n−1 |v − v∗ | dθ sinn−2 θ 2 B , cos(2θ) = |S n−2 | cos θ cos θ 0 Z π4 n−2 sin (2θ) |v − v∗ | = |S n−2 | d(2θ) B , cos(2θ) cosn θ cos θ 0 Z π2 n−2 sin θ |v − v∗ | n−2 = |S | dθ B , cos θ . cosn (θ/2) cos(θ/2) 0 This proves the first claim. For the complete proof and comments, we refer to [9]. 2n−1 dσ B (k · σ)2

5. Coercivity results. We have seen in Section 3 that one part of the Boltzmann collision operator behaves as a negative fractional Laplacian type operator, though with an extremely intricate form. Moreover, we have seen in Section 4 that the other part of this operator behaves exactly as a convolution type operator. It is thus interesting to know if Boltzmann operator induces coercivity type results, up to a non important remainder term. Since the natural setting of Boltzmann equations is based on entropic bounds, the guess is that the entropy dissipation estimate should imply such results: this is similar to the simple example of the heat equation, for which the Laplacian operator induces coercive inequalities. Of course, Boltzmann case is much more involved! This Section shows that this is the case, following Alexandre, Desvillettes, Villani and Wennberg [9] that we expose herein, following previous studies by Alexandre [1, 7, 6], Lions [95] and Villani [132]. Let us mention that there is still some works to be done, at the level of weighted estimates and at the level of the linearized dissipation functional. We mention some preliminary works due to Mouhot and coll. [23, 108, 110, 111], and we shall recall these results later on. We shall assume n ≥ 2 and typically the following lower bound on collision cross-sections, since we are interested in coercive type estimations B(v − v∗ , σ) ≥ Φ(|v − v∗ |)b(k · σ).

(51)

Above the kinetic cross-section Φ(|z|) : Rn → R+ is continuous and strictly positive for z 6= 0, and K sinn−2 θ b(cos θ) ∼ as θ → 0, ν > 0. (52) θ1+ν

BOLTZMANN EQUATION

Recall the expression of the entropy dissipation functional Z D(f ) = − Q(f, f ) log f dv n ZR Z 1 f ′ f∗′ = B(v − v∗ , σ)(f ′ f∗′ − f f∗ ) log dσ dv dv∗ . 4 R2n S n−1 f f∗

17

(53)

and the fact that there is a natural a priori bound on this quantity. Let us introduce the generalized linear (w.r.t. f ) entropy dissipation functional, for a given positive function g Z D(g, f ) = − Q(g, f ) log f. (54) Rn

For a given cross-section B, we shall still use the quantities involved in (18), (20) and (21). Theorem 5.1. [Coercivity [9]] Assume that B satisfies (51), and that b satisfies the singularity assumption (52). Furthermore, suppose that M(|z|) + |z|M′ (|z|) ≤ C0 (1 + |z|)2 . Then, for all R > 0 there exists a constant Cg,R , depending only on b, kgkL11 , kgkL log L , R, and on Φ, such that p k f k2H ν/2 (|v| 0 depends only on the deviation angle θ (Maxwellian molecules): B(v − v∗ , σ) = b(cos θ), cos θ =< with the following singularity Z

0

π 2

v − v∗ π , σ >, 0 ≤ θ ≤ . |v − v∗ | 2

sinn θ b(cos θ)dθ < +∞ and sinn−2 θ b(cos θ) ∼

κ when θ −→ 0, θ1+ν

(63)

(64)

where κ > 0 and 0 < ν < 2 are fixed. We assume, see the previous Sections, that a weak solution to Boltzmann equation (61) has already been constructed and that it satisfies the usual entropic estimate, for a fixed T > 0 (eventually T = +∞) Z sup f (t, v)(1 + |v|2 + log(1 + f (t, v))dv < ∞. (65) t∈[0,T ]

Rn

We shall also assume that this weak solution satisfies the mass conservation, but we do not use the entropy dissipation bound. The exact result we want to prove here is given by Theorem 6.1. Let be given an initial datum f0 satisfying (62) and a Maxwellian collision cross section B as in (63), with b satisfying (64). Let f be any weak non negative solution of Boltzmann spatially homogeneous equation (61), satisfying (65) and the mass conservation. Then, for any t > 0, for all s ∈ R+ , f (t, .) belongs to the Sobolev space H s (Rn ). By known results on moments propagation (see Section 1), this C ∞ regularity cannot be improved in the Maxwellian case to regularity for instance in S. 6.1. A Littlewood-Paley proof. Here is a first proof of this result. It simplifies the proof given in [10]. It was in fact important to have extremely simple proofs in order to tackle the general case of non maxwellian molecules. Arguments will use Littlewood-Paley theory, the main point being to work on small annuli instead of the full frequency space and otherwise we will only use elementary analytical arguments. But, first of all, let us recall Littlewood-Paley decomposition and some links with Sobolev type spaces, see the books of Runst, Sickel and Triebel [120, 116]. We fix once for all a collection {ψk = ψk (ξ)}k∈N of smooth functions such that supp ψ0 ⊂ {ξ ∈ RN , | ξ | ≤ 2},

and

supp ψk ⊂ {ξ ∈ RN , 2k−1 ≤ | ξ | ≤ 2k+1 } for all k ≥ 1, +∞ X

k=0

ψk (ξ) = 1 for all ξ ∈ RN .

BOLTZMANN EQUATION

21

To simplify some computations, all functions ψk , for k ≥ 1, are constructed from a single one ψ ≥ 0, i.e. we are given ψ such that supp ψ ⊂ {ξ ∈ RN , 21 ≤ | ξ | ≤ 2}, √ ψ > 0 if √12 ≤ | ξ | ≤ 2 such that ψk (ξ) ≡ ψ( 2ξk ), for all k ≥ 1 and ξ ∈ RN . Then, Littlewood-Paley projection operators pk , for k ≥ 0, are defined by ˆ pd k f (ξ) = ψk (ξ)f (ξ), yielding

f=

+∞ X

k=0

pk f for all f ∈ S ′ .

We can find a new collection {ψ˜k = ψ˜k (ξ)}k∈N of smooth functions such that supp ψ˜0 ⊂ {ξ ∈ RN , | ξ | ≤ 4}, and supp ψ˜k included in the set of ξ in RN , such that 2k−2 ≤ |ξ| ≤ 2k+2 , for all k ≥ 1, and such that ψk ψ˜k = ψk , for all integer k. As before, corresponding operator p˜k , for k ≥ 0, are defined as pd ˜k f (ξ) = ψ˜k (ξ)fˆ(ξ).

All these functions ψ˜k , for k ≥ 1, are constructed from a single one ψ˜ ≥ 0, i.e. we take ψ˜ such that supp ψ˜ ⊂ {ξ ∈ RN , 212 ≤ | ξ | ≤ 22 }, ψ˜ > 0 if 12 ≤ | ξ | ≤ 2, ˜ ξk ), for all k ≥ 1 and ξ ∈ RN . such that ψ˜k (ξ) ≡ ψ( 2 Note that, for any integer k pk p˜k = pk . (66) Moreover, using Plancherel formula, it follows that Z Z f (v)pk (resp. p˜k )g(v)dv = pk (resp. p˜k )f (v)g(v)dv, for all f, g ∈ S ′ . (67) v

v

For all f ∈ L1 , one has Bernstein’s inequality: kpk f kL2 ≤ C 2

Nk 2

kpk f kL1 , (68) ˜ where C is a constant depending on the function ψ. 1 We set for any v ∈ Rn , < v >= (1 + |v|2 ) 2 . Lebesgue weighted spaces Lpα , α ∈ R, are defined as the spaces of those functions f = f (v) such that < v > f ∈ Lp . We denote the corresponding norm by k.kLpα . Thanks to this decomposition, weighted space L1α satisfies ∞ X ∀α > 0, || f ||L1α ∼ 2js || ψj f ||L1 . j=0

More generally, usual weighted Sobolev-Besov spaces can be described by the following important result, see for instance the results quoted in the books [120, 116], last index α referring to the weight kf kqBp,q,α ≃ s

+∞ X

k=0

2jqs kpk f kqLpα ≃

+∞ X +∞ X

k=0 j=0

2kqs 2jqα kψj pk f kqLp .

We now go back to the proof of the regularization effect for Boltzmann equation. Hereafter, letter C denotes any positive constant not depending on the integer k that will appear in the proof of our result below. If we need to mention its dependence with respect to any parameter β, then we shall use the notation C(β) or Cβ .

22


Applying Fourier transform on Boltzmann equation (61), we obtain the equation, due to Bobylev [26] Z ξ \ ˆ (69) ∂t f = Q(f, f ) = [fˆ(ξ + )fˆ(ξ − ) − fˆ(ξ)fˆ(0)] b( .σ) dσ, |ξ| n−1 S

where

ξ |ξ|σ ξ |ξ|σ + and ξ − = − . 2 2 2 2 Note that this equality holds true in view of our assumptions (65) on f . Let k be an integer such that k ≥ 3. The reason for choosing this minimal value of k will become clear below. It follows from (69) that Z  2  ∂ ||p f || = Q(f, pk f )pk f dv 2  t k L   Rn (70) Z Z  +  ξ ξ ξ  + − ˆ ˆ d  +2Re pk f (ξ)f (ξ )f (ξ )][ψ( k ) − ψ( k )]b( .σ)dσdξ. 2 2 |ξ| Rn S n−1 ξ+ =

Using the change of variables (v, v∗ , σ) 7→ (v ′ , v∗′ , −σ), the first term on the right hand side of (70) becomes Z Z v − v∗ Q(f, pk f )pk f dv = f∗ pk f [(pk f )′ − pk f ]b( .σ)dv∗ dvdσ |v − v∗ | Rn Rn ×Rn ×S n−1 Using the simple equality

pk f [(pk f )′ − pk f ] =

1 1 [(pk f )′2 − pk f 2 ] − [(pk f )′ − pk f ]2 , 2 2

(70) finally becomes R  v−v∗ ∂t ||pk f ||2L2 + Rn ×Rn ×S n−1 f∗ [(pk f )′ − pk f ]2 b( |v−v .σ)dv∗ dvdσ  ∗|      R R ξ ξ+ ξ ˆ + ˆ − = 2Re Rn S n−1 pd k f (ξ)f (ξ )f (ξ )][ψ( 2k ) − ψ( 2k )]b( |ξ| .σ)dσdξ       +R v−v∗ ′2 2 Rn ×Rn ×S n−1 f∗ [(pk f ) − pk f ]b( |v−v∗ | .σ)dv∗ dvdσ.

(71)

For the second term on the l.h.s. of (71), we use the coercivity estimate from the previous Section 5 to get ZZZ Z v − v∗ ′ 2 2 ν kν 2 f∗ [(pk f ) − pk f ] b( .σ)dv∗ dvdσ & |pd k f | |ξ| dξ & 2 ||pk f ||L2 . |v − v∗ | Rn

Concerning the second term on the r.h.s. of (71), we use this time the Cancellation Lemma 4.1 (exchanging v into v∗ ), to get ZZZ v − v∗ f∗ [(pk f )′2 − pk f 2 ]b( .σ)dv∗ dvdσ . ||pk f ||2L2 |v − v∗ | Finally, there remains to estimate the first term on the r.h.s. of (71). This integral is nothing else than Z Z ˆ + ˆ − k ξ .σ)dσdξ, 2Re pd k f (ξ)f (ξ )f (ξ )Aξ b( |ξ| Rn S n−1

where

Akξ = ψ(

ξ ξ+ ) − ψ( ). 2k 2k

(72)

BOLTZMANN EQUATION

23

Thus (72) is bounded from above by Z Z ξ k b ˆ + 2f0 (0) |pd .σ)dσdξ. k f (ξ)| |f (ξ )| |Aξ | b( |ξ| Rn S n−1

(73)

Let us notice that for 2k−1 ≤ | ξ | ≤ 2k+1 , it follows that 2k−2 ≤ | ξ + | ≤ 2k+1 . Thus + d + d \ + \ + fˆ(ξ + ) = p\ ˜k f (ξ + ), k−2 f (ξ ) + pk−1 f (ξ ) + pk f (ξ ) + pk+1 f (ξ ) = p

for another Littlewood Paley decomposition p˜k . On the other hand, we can also easily estimate |Akξ | as follows

θ |Akξ | . sin2 . 2 Thus, we are left to estimate from above Z Z ξ d Hk ≡ 2fb0 (0) |pd ˜k f (ξ + )||Akξ |b( .σ)dσdξ, k f (ξ)||p |ξ| 2k−1 ≤|ξ|≤2k+1 S n−1

(74)

(75)

that we bound as where

and

Mk ≡ fb0 (0) Nk ≡ fb0 (0)

Hk ≤ M k + N k ,

Z

2k−1 ≤|ξ|≤2k+1

Z

2k−1 ≤|ξ|≤2k+1

Z

S n−1

Z

S n−1

2 k |pd k f (ξ)| |Aξ |b(

ξ .σ)dσdξ |ξ|

(76)

ξ .σ)dσdξ. |ξ|

(77)

|pd ˜k f (ξ + )|2 |Akξ |b(

Clearly, in view of estimate (74), one has

Mk . fb0 (0)||pk (f )||2L2 .

As for (77), using (74) and the change of variable ξ + 7→ ξ (see for example the proof of the Cancellation Lemma 4.1 in [9]), we get

All in all, we have obtained

Nk . fb0 (0)||˜ pk (f )||2L2 .

Hk . fb0 (0){||pk (f )||2L2 + ||˜ pk (f )||2L2 }.

Gluing all the above estimates, we find that Z 2 ν 2 ˆ ∂t ||pk f ||2L2 + C(f0 ) |pd pk (f )||2L2 }, k f | |ξ| dξ ≤ C f0 (0){||pk f ||L2 + ||˜ Rn

k−1

and using the fact that 2 ≤ |ξ| ≤ 2k+1 in the integral of the second term in the left hand side, we obtain (for another constant still denoted by C(f0 )), absorbing an unimportant term on the right hand side, for k big enough ∂t ||pk f ||2L2 + C(f0 ) 2kν ||pk f ||2L2 ≤ C||˜ pk (f )||2L2 .

(78)

˜ nk ||pk f ||2 1 . Since ||pk f ||2 1 ≤ Bernstein’s inequality yields ||pk f ||2L2 ≤ C(ψ)2 L L 2 2 ˜ ˜ C(ψ)||f ||L1 = C(ψ)||f0 ||L1 , clearly for all t ≥ 0, ||pk f ||2L2 . 2nk . Then, using (78), nk one obtains that for t > t0 > 0, ||pk f ||2L2 . 22kν , and by iterating also on the time interval, for all s ≥ 0, for all t > 0, ||pk f ||2L2 .

2nk 2ksν

, which ends the proof.

24


6.2. Further improvements. The above proof uses Littlewood Paley theory and simplifies the proof published in [10]. A few years later, Morimoto and coll. [105] while studying Gevrey regularity, produced a different argument, avoiding the use of Littlewood Paley theory. Their key idea is the following, and we restrict here for the exposition to velocity dimension 3. First of all, we note that f (t) ∈ L1 ⊂ H −2 . Then for t ≥ 0, N > 0, and 0 < δ < 1, we introduce the multiplicator Mδ (t, ξ) = (1 + |ξ|2 )

N t−4 2

.(1 + δ|ξ|2 )−N0

with N0 = N2T + 2. Then for any t ∈ t ∈ (0, T ) and δ ∈]0, 1[, we note that Mδ (t, Dv )f ∈ W 2,∞ . The idea is then to use M 2 (v, Dv )f as a test function in the weak formulation associated with Boltzmann equation. We need to commute with the nonlinear Boltzmann operator, and its action is given by the following Lemma borrowed from [105] and whose proof mimics some arguments which were somehow hidden in the previous proof of [10] Lemma 6.2. One has |(Q(f, Mδ f ), Mδ f )L2 − (Q(f, f ), Mδ2 f )L2 | . ||Mδ f ||2L2 . With the help of this commutation lemma, using the previous arguments in order to use both the Coercivity estimate and the Cancellation Lemma, one obtains the final estimation ||Mδ f (t)||2L2 . eCt ||Mδ f (0)||2L2 and thus by taking N sufficiently large, we get the regularisation property. Let us make some final comments as regards this Section: it is also shown in [105] that we can take into account logarithmic type Coulomb Maxwellian kernels. That assumptions uses then a slightly modified coercivity estimate in Log-Sobolev space, but see also the comments in [9]. Moreover, Gevrey regularity is also shown in [105] for the linear spatially homogeneous Boltzmann equation, still in the Maxwellian case. 7. Functional estimates for the collision operator. After these extremely simple proofs of regularization properties in the case of the spatially homogeneous Boltzmann equation for Maxwellian molecules, the question was therefore to study more general cross sections. It soon appears that commutators estimates were needed (it is not really needed explicitly in the Maxwellian molecules case), and thus a more precise study of the full Boltzmann operator appears to be necessary (but is it really so?). Therefore, the next step was to produce more precise functional properties on Boltzmann collision operator Q, for any dimension n ≥ 2, and this has been done in [8, 12], and then for the particular case of L2 type estimates, proven differently in [14, 105], for a restricted class of cross sections. We shall first present the results from [8, 12], without proofs, and sketch a variant of proof in the particular case of L2 type spaces in the spirit of [14, 105]. 7.1. Besov type estimates. We still assume that B(v − v∗ , σ) is supported in the set(0 ≤ θ ≤ π), i.e, (v − v∗ , σ) ≥ 0.

BOLTZMANN EQUATION

25

Let us recall that the typical example is given by inverse s-power repulsive forces in the dimension n = 3, in which case the kernel B is given by s−5 B(v − v∗ , σ) = |v − v∗ |γs b(~k · σ) with γs = . s−1 sin θb(cos θ) ∼ Kθ−1−νs

as θ → 0,

with νs =

2 > 0, K > 0. s−1

(79)

(80)

Above, s > 2, so that 0 < νs < 2 and −3 < γs < 1. Note also that since γs = 1 − 2νs , it follows that −1 < νs + γs < 0, if 1 < νs < 2. We are interested in the following bilinear operator Z Z Q(g, f ) = dv∗ dσB(v − v∗ , σ)(g∗′ f ′ − g∗ f ). (81) Rn

S n−1

We shall make the following assumptions on B   B(v − v∗ , σ) = Φ(|v − v∗ |)b(cos θ), sinn−2 θb(cos θ) ∼ Kθ−1−ν , ν > 0, K > 0, Φ(|z|) . |z|γ for |z| . 1 and Φ(|z|) . |z|β for |z| & 1,  γ > −n, β ∈ R, 1 ≤ ν < 2. (82) The first result uses a very particular scale of Besov spaces and is relatively the easiest one to obtain Theorem 7.1. Let Assumption (82) holds true. Then, for any sufficiently smooth functions f and g: 1. if β + ν ≤ 0 and γ + ν ≥ 1, one has kQ(g, f )kb−ν . kgkL1 kf kL1 . 1,∞

2. if β + ν ≥ 0 and γ + ν ≥ 1, one has for β ′ ≥ β + ν, kQ(g, f )kb−ν . kgkL1 ′ kf kL1 ′ . 1,∞

β

β

The second result uses the traditional scale of Sobolev spaces and more generally s of (weighted) Besov Bp,q , see in particular [118, 119, 120], but covers the specific Maxwellian case, that is, the case when Φ(|v|) is constant valued. This is a special case of Assumption (82) above, corresponding to γ = β = 0. Theorem 7.2. (Maxwellian case) Assume that Φ(|v|) = 1. Then Q = Qmax satisfies the following estimate, for all s ∈ R, all 1 ≤ p , q < ∞, and for any sufficiently smooth functions f and g: s−ν . kgkL1 kf k s−ν . kQmax (g, f )kBp,q Bp,q,ν ν

Note carefully that, in this last result, no regularity on g is required, except for a moment estimate. More precisely, we need only g to belong to L1ν . This point was noticed in [13], and in fact follows very easily by looking carefully to the proof of the corresponding result in [8, 12]. In particular, this point is linked with a special Parseval type formula which holds true for the Maxwellian case. With this special Maxwellian case, a last main result from [8, 12] considers smoothed versions of kernels close to |v|ν , where we simplify the framework by taking β = γ.

26


Theorem 7.3. Assume that Φ(v) = c < v >γ for some γ < ν and some positive constant c. Then, for all s ∈ R, all 1 ≤ p , q < ∞, for any sufficiently smooth functions f and g, one has s−ν . kgkL1 kQ(g, f )kBp,q

ν+γ +

kf kB s

p,q,ν+γ +

.

The proof of this result uses the Maxwellian case together with some commutators estimates. Let us also note that it holds if Φ belongs to S γ (standard notations from pseudo-differential operator theory). We shall not give the proofs of these results, but just say that the analysis of the bilinear Boltzmann operator (81) is based on a detailed study of the linear operator, for a given v∗ , Z TvΦ∗ : f 7→ B(v − v ∗ , σ)(f ′ − f )dσ. (83) S n−1

which we have already seen in action previously. Noticing that TvΦ∗ = τ−v∗ ◦ T Φ ◦ τv∗ , where τv∗ denotes the usual translation operator along vector v∗ and T Φ denotes the operator Z v ± |v|σ Φ T f (v) ≡ B(v, σ){(f (v + ) − f (v))}dσ, v ± ≡ , (84) 2 S n−1

it is therefore enough, at least in translation invariant functional spaces, to study this last operator. Then, we use a Littlewood-Paley decomposition and other well-known results from harmonic analysis [118, 119, 120], together with a detailed study of commutator type estimates. For all detailed proofs, let us refer once again to the original papers [8, 12]. Note by the way that some of the proofs therein may be considerably shorten.

7.2. A simple proof. Instead, we now present a result in the hilbertian framework, building upon ideas taken from [14, 105]. The proof is sufficiently short so that we have decided to include it, and it will give some of the ideas of this whole section. We specialize to velocity dimension 3, but there is really nothing special about this dimension. We want to prove γ

Theorem 7.4. Assume that B(v − v∗ , σ) = Φ(v − v∗ )b(cos θ), with Φ(z) =< z > 2 κ and sin θb(cos θ) ∼ θ1+ν when θ → 0, for some κ > 0 and 0 < ν < 2 fixed, and γ ∈ R. Then for any m ∈ R , one has kQ(f, g)kH m (R3v ) ≤ Ckf kL1

(R3v ) (γ+ν)+

kgkH m+ν

(R3v ) (γ+ν)+

.

(85)

Remark 7.5. Let us mention that by looking carefully to the proof below, the singular case Φ(z) = |z|γ can be also considered. Furthermore, at least for smoothed cross-sections, one can certainly avoid any use of Littlewood-Paley decompositions. We start the proof by writing < Q(f, g); h >=

Z

R3

=

dv∗ f∗

Z

R3

nZ

dv∗ f∗

dv

R3

Z

R3

Z

S2

dσb(

o v − v∗ .σ)Φ(v − v∗ )g(v)[h′ − h] |v − v∗ |

dvg(v)τ−v∗ ◦ T Φ ◦ τv∗ h(v),

BOLTZMANN EQUATION

27

where, again, for any smooth function f , T Φ (f ) is defined by Z v Φ T (f )(v) = dσb( .σ)Φ(v)[f (v + ) − f (v)]. |v| 2 S

Thus it is enough to study T Φ . Using a Littlewook Paley decomposition (this time, on the velocity side!), one has Z XZ dvT Φ (f )(v).g(v) = dvφk (v)T Φ (f )(v)φk (v)g(v), R3

R3

k

and introducing another Littlewod Paley decomposition φ˜k , Z XZ Φ dvT (f )(v).g(v) = dvT Φk (φ˜k f ).φk (v)g(v). R3

R3

k

Here Φk = φk Φ. Setting fk = φ˜k f and gk = φk g, it follows that Z XZ Φ dvT (f )(v).g(v) = dvT Φk (fk )(v)gk (v). R3

R3

k

Thus, we need to study, for each k, the term Z dvT Φk (fk )(v)gk (v). R3

Φk

We know that T is the adjoint to the operator f → QΦk (δ0 , f ), see for example the comments in [9, 16, 134]. Thus Z Z Z k (δ , g )(ξ), dvT Φk (fk )(v)gk (v) = dvfk QΦk (δ0 , gk ) = dξ fbk (ξ)QΦ\ 0 k R3

R3

R3

and an application of Bobylev’s ideas (see the Appendix of [9]) tells us that k (δ , g ) QΦ\ 0 k

=

Z

dξ∗

R3

which is also

Z

ck (|ξ∗ |)b( dσ Φ

S2

Z

n o ξ .σ) gˆk (ξ + − ξ∗ ) − gˆk (ξ − ξ∗ ) . |ξ|

ξ + [ [ .σ)[Φ k gk (ξ ) − Φk gk (ξ)]. |ξ| S2 Setting Gk = Φk gk , all in all, we have obtained k (δ , g ) = QΦ\ 0 k

Z

Φ

dvT (f )(v).g(v) =

R3

XZ k

dσb(

dξ fbk (ξ).

R3

We split Ak as

Z

dσb( S2

X ξ ck (ξ + ) − G ck (ξ)] ≡ .σ)[G Ak . |ξ| k (87)

Ak = A1,k + A2,k where A1,k =

Z

dξ fbk (ξ).

R3

and A2,k =

Z

dξ fbk (ξ).

R3

Z

dσ1sin θ ≤2−k −1 b(

Z

dσ1sin θ ≥2−k −1 b(

S2

S2

2

2

(86)

ξ ck (ξ + ) − G ck (ξ)] .σ)[G |ξ|

ξ ck (ξ + ) − G ck (ξ)]. .σ)[G |ξ|

28


For the analysis of A1,k , we use Taylor formula at order two, as in [16], ck (ξ + ) − G ck (ξ) = −∇ξ G ck (ξ).ξ − − G

and thus

Z

0

1

ck (ξ + + τ ξ − ).ξ − .ξ − , dτ Dξ2 G

A1,k = A1,k,1 + A1,k,2

where A1,k,1 = − and

Z

dξ fbk (ξ).

R3

Z


S2

2

ξ ck (ξ).ξ − ] .σ)[∇ξ G |ξ|

Z 1 ξ ck (ξ + + τ ξ − ).ξ − .ξ − ]. .σ)[ dτ Dξ2 G 2 |ξ| 3 2 R S 0 ξ For the analysis of A1,k,1 , we note first that 1sin θ ≤2−k −1 b( .σ) is a function 2 |ξ| ξ of cos θ and |ξ| (depending on k) where cos θ = |ξ| .σ. Denote it by B k (cos θ, ξ). Set ξ also ξ˜ = |ξ| . Then by the symmetry argument of [16], one has Z Z ˜ ξ. ˜ dσdσB k (cos θ, ξ)ξ − = dσdσB k (cos θ, ξ)[ξ − .ξ] A1,k,2 = −

Z

dξ fbk (ξ).

Z


S2

S2

˜ ξ˜ = − 1 |ξ|(1 − cos θ)ξ˜ = − sin2 θ ξ. Thus, it follows that On the other hand [ξ .ξ] 2 2 Z Z ˜ ξ˜ dσdσB k (cos θ, ξ)ξ − = dσdσB k (cos θ, ξ)[ξ − .ξ] 2 S2 S Z θ =− dσdσB k (cos θ, ξ) sin2 ξ. 2 S2 Therefore, we have found that Z Z θ ξ b ck (ξ).ξ]. A1,k,1 = dξ fk (ξ). dσ1sin θ ≤2−k −1 b( .σ) sin2 [∇ξ G 2 |ξ| 2 3 2 R S ξ θ Notice that b( .σ) sin2 is integrable and an explicit computation shows you |ξ| 2 also that, since sin θ b(cos θ).θ2 ∼ θ1−ν as θ → 0 Z ξ θ dσdσ1sin θ ≤2−k −1 b( .σ) sin2 . 2k(ν−2) < ξ >ν−2 . 2 |ξ| 2 2 S Therefore Z ck (ξ)| |A1,k,1 | . dξ fbk (ξ).2k(ν−2) < ξ >ν−1 |∇ξ G 3 R Z ck (ξ)| . 2k(ν−2) dξ fbk (ξ) < ξ >m < ξ >ν−1 < ξ >−m |∇ξ G −

R3

and using Cauchy Schwartz inequality, it follows that Z 1 k(ν−2) |A1,k,1 | . 2 { dξ|fbk (ξ)|2 < ξ >2(m+ν−1) dξ} 2 . 3 R Z ck (ξ)|2 } 12 ×{ dξ < ξ >−2m |∇ξ G R3

BOLTZMANN EQUATION

29

that is ck (ξ)||L2 . |A1,k,1 | . 2k(ν−2) || < Dv >m+ν−1 fk ||L2 || < ξ >−m |∇ξ G This is not enough, since we need to clear out the weight. For this purpose, note ck (ξ) = ∇ξ [Φ ˆ k ∗ gbk ] = [∇ξ Φ ˆ k ] ∗ gbk , and recall that that ∇ξ G Z v c Φk (ξ) = < v >γ φ( k )e−iv.ξ dv. 2 R3 Thus, if k ≥ 1 (for a function φ supported in an annulus far away from 0) Z Z k v ck (ξ) = −i ∇ξ Φ < v >γ φ( k )ve−iv.ξ dv = −i24k < 2k v >γ φ(v)ve−iv.2 ξ dv. 2 R3 R3

ck (ξ)| . 24k 2kγ . This One can then shows that, for any n ≥ 0, < 2k ξ >n |∇ξ Φ holds true for any value of γ ∈ R, with k ≥ 1. But when k = 0, the function φ has then its support in a small ball around 0, and then the above estimate also holds true. Then, we first note that, using Petree’s inequality Z ck (ξ)| = | < ξ >−m ck (ξ∗ )gbk (ξ − ξ∗ )| | < ξ >−m ∇ξ G dξ∗ ∇ξ Φ R3 Z ck (ξ∗ ) < ξ∗ >|m| ||gbk (ξ − ξ∗ )| < ξ − ξ∗ >−m ≤ Cm dξ∗ |∇ξ Φ R3

which is again in the form of a convolution. Thus, using Holder inequality for convolution, it follows that ck (ξ)||L2 . ||∇ξ Φ ck (ξ∗ ) < ξ∗ >|m| ||L1 .|| < ξ >−m |gbk (ξ)||L2 . || < ξ >−m ∇ξ G

Since < ξ∗ >|m| ≤< 2k ξ∗ >|m| , by using the above punctual estimate for large n, it follows that we can cancel the exponent 4 (recall we are in 3d) and get ck (ξ)||L2 . 2k 2kγ || < ξ >−m |gbk (ξ)||L2 . || < ξ >−m ∇ξ G

Finally, it follows that

|A1,k,1 | . 2k(ν−2) || < Dv >m+ν−1 fk ||L2 .2k 2kγ || < Dv >−m gk ||L2 . 2k(ν+γ−1) || < Dv >m+ν−1 fk ||L2 || < Dv >−m gk ||L2 . Turning to the term A1,k,2 , one has |A1,k,2 | .

Z 1Z Z ξ ck (ξ + + τ (ξ − ))|.|ξ − |2 |fbk (ξ)|dξdσdτ [ 1sin θ ≤2−k −1 b( .σ).|Dξ2 G 2 |ξ| 0 R3 S 2

Z 1Z Z θ ck (ξ + + τ (ξ − ))||fbk (ξ)|dξdσdτ. [ 1sin θ ≤2−k −1 b(.) sin2 < ξ >2 |Dξ2 G 2 2 3 2 0 R S Setting ˜b ξ .σ = b ξ .σ sin2 θ , |ξ| |ξ| 2 ξ ˜ then b is of course still a function of .σ, and for any value of ν, is integrable over .

S 2 . Since again

|ξ|

30


Z

ξ 1sin θ ≤2−k −1 ˜b( .σ) . 2k(ν−2) < ξ >ν−2 , 2 |ξ| σ we can use Cauchy Schwartz inequality to get, writing < ξ >2 =< ξ >2α . < ξ >2β , with α + β = 1, for any m ∈ R |A1,k,2 | . Z 1Z Z ξ ck (ξ + + τ (ξ − ))|2 dσdξdτ ] 12 1sin θ ≤2−k −1 ˜b( .σ) < ξ >4α < ξ >−2m |Dξ2 G [ 2 |ξ| 0 R3 S 2 Z 1Z Z 1 ξ ×[ 1sin θ ≤2−k −1 ˜b( .σ)|fbk (ξ)|2 < ξ >4β < ξ >2m dξdσdτ ] 2 2 |ξ| 0 R3 S 2 1

1

. A2 × B 2 For B, fixing ξ and integrating w.r.t. σ with pole direction ξ/|ξ|, we get, Z B. |fbk (ξ)|2 < ξ >4β < ξ >2m < ξ >ν−2 2k(ν−2) dξ, R3

and by choosing β such that 4β + ν − 2 = 2ν, that is 2β + β = 12 (1 + ν2 ), we get 1

ν 2

− 1 = ν, that is

ν

B . 2k(ν−2) || < Dv >m+ν fk ||2L2 and thus B 2 . 2k( 2 −1) || < Dv >m+ν fk ||L2 .

Note that α = 12 (1 − ν2 ). For the analysis of A, we want to perform the change of variables for fixed τ ∈ [0, 1], σ ∈ S 2 , which thus generalises the one introduced in [9], see also Section 4: ξ → ξτ = ξ + + τ ξ − . ξ Its Jacobian is given by, for k = |ξ| |

∂(ξτ ) 1+τ 1−τ | = |det( I+ σ ⊗ k)| ∂(v) 2 2

(1 + τ )3 1−τ (1 + τ )3 2τ 1−τ θ |1 + k.σ| = | +2 cos2 | 3 3 2 1+τ 2 1+τ 1+τ 2 (1 + τ )3 2τ 1−τ (1 + τ )3 1 ≥ | + |= ≥ 3 23 1+τ 1+τ 23 2 and is therefore bounded from below uniformly w.r.t. to σ and τ . Since 1 1 ξ + = [ξ + |ξ|σ] and ξ − = [ξ − |ξ|σ], 2 2 we note that, taking the square of the norms, and the scalar product with σ the following formula =

and

θ θ θ ξ+ |ξ + | = |ξ| cos , ξ + .σ = |ξ| cos2 , cos = + .σ 2 2 2 |ξ |

θ θ θ ξ− |ξ − | = |ξ| sin , ξ − .σ = |ξ| sin2 , sin = − .σ. 2 2 2 |ξ | Since (because ξ + .ξ − = 0) ξτ = ξ + + τ ξ − , we get

BOLTZMANN EQUATION

31

θ θ θ + τ 2 sin2 } = |ξ|2 {(1 − τ 2 ) cos2 + τ 2 } 2 2 2

|ξτ |2 = |ξ + |2 + τ 2 |ξ − |2 = |ξ|2 {cos2 and

θ θ θ + τ sin2 } = |ξ|{(1 − τ ) cos2 + τ }. 2 2 2 Dividing these two relations, we get in particular ξτ .σ = ξ + .σ + τ ξ − .σ = |ξ|{cos2

{cos2 (ξτ .σ)2 = |ξτ |2 {cos2 and since τ 2 ≤ τ , we see that

(ξτ .σ)2 |ξτ |2

θ 2 θ 2

+ τ sin2 θ2 }2

+ τ 2 sin2 θ2 }

,

≥ cos2 θ2 . Since our ˜b writes as

˜b(cos θ) = ˜b(2 cos2 θ − 1), 2 we can assume w.l.o.g. that our ˜b is increasing with respect to cos θ =

ξ |ξ| .σ,

thus

2 ˜b(cos θ) = ˜b(2 cos2 θ − 1) ≤ ˜b(2 (ξτ .σ) − 1). 2 |ξτ |2

Thus, in analysing A, we can use this upper bound, which simplifies the change of variables ξ → ξτ . On the other hand, we must take into account the set sin θ2 ≤ 2−k < ξ >−1 . Since < ξ >≥< ξτ >, this set is included in the set sin θ2 ≤ 2−k < ξτ >−1 , which is the same set as cos2 θ2 ≥ 1 − 2−2k < ξτ >−2 . But this last set is included in the set (ξτ .σ)2 ≥ 1 − 2−2k < ξτ >−2 . |ξτ |2

Thus when we change variables, it suffices to introduce the angle cos α2 = so the above set is exactly similar but in the new angles!! All in all, we have, noticing also that |ξτ | ≤ |ξ| ≤ c|ξτ |, A.

Z

Z [

ξτ .σ |ξτ | ,

ξ ck (ξ))|2 dσdξdτ Isin θ ≤2−k −1 ˜b( .σ) < ξ >4α < ξ >−2m |Dξ2 G 2 |ξ| S 2 R3 Z ck (ξ))|2 dσdξdτ. . 2k(ν−2) < ξ >ν−2 < ξ >4α < ξ >−2m |Dξ2 G R3

Since ν − 2 + 4α = 0, it follows that Z ck (ξ))|2 dσdξdτ. A . 2k(ν−2) < ξ >−2m |Dξ2 G R3

Thus

1

ν

ck (ξ)||L2 A 2 . 2k( 2 −1) || < ξ >−m |Dξ2 G

ν

ν

. 2k( 2 −1) 22k 2kγ || < Dv >−m gk ||L2 . 2k( 2 +1) 2kγ || < Dv >−m gk ||L2

Thus, all in all, we have ν

ν

|A1,k,2 | . 2k( 2 +1) 2kγ || < Dv >−m gk ||L2 .2k( 2 −1) || < Dv >m+ν fk ||L2

32


. 2k(ν+γ) || < Dv >−m gk ||L2 .|| < Dv >m+ν fk ||L2 . We can now conclude for A1,k . Taking into account the estimations for A1,k,1 and for A1,k,2 , we have arrived at A1,k . 2k(ν+γ) || < Dv >−m gk ||L2 || < Dv >m+ν fk ||L2 . For the analysis of A2,k , by definition, one has Z Z ξ ck (ξ + ) − G ck (ξ)]. A2,k = dξ fbk (ξ). dσ1sin θ ≥2−k −1 b( .σ)[G 2 |ξ| 3 2 R S Here we are in a situation similar to the cutoff case. First we note that Z ξ b( .σ)1sin θ ≥2−k −1 dσ .< ξ >ν 2kν . 2 |ξ| S2 Then one has Z Z ck (ξ + )| |A2,k | ≤ dξ dσb(ξ/|ξ|.σ)1sin θ ≥2−k −1 |fbk (ξ)||G 2 R3 S2 Z Z ck (ξ)| + dξ dσb(ξ/|ξ|.σ)1sin θ ≥2−k −1 |fbk (ξ)||G R3

2

S2

≤ I + II. Since II is similar to I (except that we do not need any change of variables), we just concentrate on I. One has Z Z ck (ξ + )| I= dξ dσb(ξ/|ξ|.σ)1sin θ ≥2−k −1 |fbk (ξ)||G 2 R3 S2 ZZ ξ = dξdσb( .σ)1sin θ ≥2−k −1 . 2 |ξ| m ν b −m ck (ξ + )|. × < ξ > < ξ > |fk (ξ)| < ξ > < ξ >−ν |G

By using Cauchy Schartz inequality, then the change of variables ξ → ξ + which does not modify substantially the integration set as shown above (case τ = 0), taking into account the above spherical estimate, we get kν

I.2 {

Z

dξ < ξ >

R3

2(m+ν)

1 |fbk (ξ)|2 } 2 × {

. 2kν || < Dv >m+ν fk ||L2 {

Z

Z

ck (ξ)|2 } 12 dξ < ξ >−2m |G

R3

ck (ξ)|2 } 21 . dξ < ξ >−2m |G

R3

For the last integral, this is similar to previous computations: we find a power of k less than one (because there is no ∇ξ Φk ) I . 2k(ν+γ) || < Dv >m+ν fk ||L2 || < Dv >−m gk ||L2 . Thus, since II is also similar, we have obtained |A2,k | . 2k(ν+γ) || < Dv >m+ν fk ||L2 || < Dv >−m gk ||L2 . We may now conclude: it follows that |Ak | . 2k(ν+γ) || < Dv >m+ν fk ||L2 || < Dv >−m gk ||L2 .

BOLTZMANN EQUATION

Thus | .

X k

. {2

Z

R3

dvT Φ (f )(v)g(v)| = |

X k

33

Ak |

2k(ν+γ) || < Dv >m+ν fk ||L2 || < Dv >−m gk ||L2

2k(ν+γ)

1

|| < Dv >m+ν fk ||2L2 } 2 {

X k

1

|| < Dv >−m gk ||2L2 } 2

. ||f ||H m+ν ||g||H −m ν+γ

and similarly | Thus

Z

R3

dvT Φ (f )(v)g(v)| . ||f ||H m+ν ||g||H −m

ν+γ

||T Φ (f )||H m . ||f ||H m+ν , ν+γ

and also m ||T Φ (f )||H−(ν+γ) . ||f ||H m+ν

This estimate is enough to prove the Theorem, since it only involves translation of weights, and also explains why a weight appears on f . Indeed, Z Z | < Q(f, g); h > | = | dv∗ f∗ dvg(v)τ−v∗ ◦ T Φ ◦ τv∗ h(v)| R3 R3 Z Z ≤ dv∗ |f∗ || dvg(v)τ−v∗ ◦ T Φ ◦ τv∗ h(v)| R3 R3 Z ≤ dv∗ |f∗ | ||g||H m+ν ||τ−v∗ ◦ T Φ ◦ τv∗ h||H −(m+ν) ν+γ −(ν+γ) 3 R Z . dv∗ |f∗ | < v∗ >ν+γ ||g||H m+ν ||h||H −m . ||f ||L1ν+γ ||g||H m+ν ||h||H −m R3

ν+γ

ν+γ

which gives the Theorem.

Remark 7.6. It might be possible by using similar ideas to recover the right Lebesgue exponent (that is L1 instead of L2 ) in the regularity of the gain term in the papers [28, 103], see the comments and results in [91, 92, 91, 112, 140]. 8. Spatially homogeneous Boltzmann equation: Non Maxwellian case. After these functional results on the Boltzmann collision operator, we wish to extend the regularization properties shown in the Maxwellian case, and consider a larger class of collision sections, especially those corresponding to so called hard potentials. The initial datum f0 6= 0 is again supposed to satisfy the usual “entropic” bounds, and we still assume that the collision cross section B(v − v∗ , σ) > 0 is given under the following multiplicative form v − v∗ π B(v − v∗ , σ) = Φ(| v − v∗ |)b(cos θ), cos θ =< , σ >, 0 ≤ θ ≤ . (88) |v − v∗ | 2 with the non cutoff assumption Z π2 sinn θ b(cos θ)dθ < +∞ and sinn−2 θ b(cos θ) ∼ 0

κ θ1+ν

when θ −→ 0,

(89)

34


κ > 0 and 0 < ν < 1 being two fixed constants. Function Φ, shall be assumed to correspond to a smoothed version of the so called hard potentials, that is γ

Φ(|v|) = (1 + |v|2 ) 2 ,

(90)

with the range of parameters 0 < γ ≤ 1; the real hard potentials case corresponds to the case Φ(|v|) = |v|γ . We assume that a weak solution to Boltzmann equation has already been constructed and that it satisfies the usual entropic estimate, for a fixed T > 0 (eventually T = +∞), together with mass conservation, decrease of energy and entropy dissipation rate bounded, though we shall not use this last condition. Such a weak solution shall be referred to as en entropic weak solution. It is then known that such a weak solution has then all higher moments with respect to velocity, for strictly positive time, see the recalled results from Section 2. Though the proof sketched for the Maxwellian molecules case was extremely simple, it seems not to work for non maxwellian molecules. In this non maxwellian and non cutoff case, the first results about this regularization property question were due to Desvillettes and Wennberg [56], showing S (in fact through weighted Sobolev spaces) regularity. In fact, Desvillettes and Wennberg show that, under suitable assumptions on the cross section, a solution in S does exist, with an entropic initial data f0 . Here again, we wish to show the stronger result that any entropic weak solution is smooth. Thus, this type of result is strictly not comparable to [56], except if we have an uniqueness result to conclude. A similar result was established in the context of Landau spatially homogeneous equation [61], see later on. By now, there are two different proofs of this type of property, the first one being [11], still using Littlewood Paley arguments together with the results on upper bounds from the study of the collision operator, and then [87], adding pseudodifferential calculus. [87] is the most complete work up to now and is technically more involved. In any case, one point needs to be removed: the smoothness assumption made on the kinetic part, close to null relative velocities. 8.1. A first conditional result. We shall take a few lines to sketch the proof of [11], still based on Littlewood-Paley theory. But, we need here commutators estimates, to take into account the fact that Φ is now really a non constant function. One important hypothesis which was made in [11] was to assume that for some t0 > 0, f ∈ L∞ ([t0 , +∞); L2α (Rn )), for all α ∈ R, (91)

where L2α (Rn ) denotes weighted Lebesgue space. This assumption is compatible with [52]. But note that it is still an open problem to show that any entropic weak solution (thus satisfying only the usual entropic bounds) enjoys automatically this L2 integrability (91), even if we do take t0 = 0 and an initial datum in the same class. This is in particuliar due to the lack of a good uniqueness result, and also to the fact that power of f , for an exponent less than 1 belongs to a worse Besov type space, see [116] for instance, and in view of the (quite optimal with respect to the index of regularity) functional properties of Boltzmann operator mentioned previously. However, in some cases, we can avoid making Assumption (91), see the comments later on.

BOLTZMANN EQUATION

35

Let us mention that the weighted space L2α appearing in Assumption (91) can be replaced by any weighted space Lpα , but with p > 1, though the natural case should be p = 1. Moreover, only a finite number of moments is really needed. However, Assumption (91) enables to give simple proofs. Note that this way mimics partially the strategy of [54]. One has Theorem 8.1 ([11]). . Let be given an initial datum f0 and a collision cross section B such that (88), (89) and (90) hold true. Let f be any entropic weak non negative solution of Boltzmann spatially homogeneous equation. Furthermore, we assume that (91) holds true for some t0 > 0. Then, for any t > t0 , for all s, α ∈ R+ , f (t, .) s belongs to the weighted Besov-Sobolev space B2,2,α (Rn ). In particular, it follows that for t > t0 , f (t) belongs immediately to Schwartz space S(Rn ). Proof. We just explain the main ideas. We again use the Littlewood Paley decomposition functions ψj , but also for the moments control, and associated operators pk , for any j ∈ N, k ∈ N. Then, one has, < .; . > denoting the usual duality bracket, Z Z < ψj pk Q(g, f ); ψj pk f >= dv∗ g∗ dv f τ−v∗ ◦ T Φ ◦ τv∗ [pk ψj2 pk f ], (92) Rn

Rn

where, we have introduced g = f in order to show clearly on which functions we are going to perform fractional differentiation. Furthermore, as in Section 7, for any suitable test function φ, operator T Φ is defined by Z v T Φ φ(v) = [φ(v + ) − φ(v)]b( .σ)dσ Φ(v), |v| S n−1

τv∗ denoting the usual translation, and for fixed v ∈ Rn , σ ∈ S n−1 , v± = v±|v|σ . 2 When Φ ≡ 1, corresponding to the Maxwellian case, note that, see [9] for instance, v 7→ T 1 φ(v) is adjoint to f 7→ QMax (δv∗ =0 , f ), QMax denoting Boltzmann operator in the case of Maxwellian molecules. We then write, [., .] standing for the usual commutator of two operators, where

and

< ψj pk Q(g, f ); ψj pk f >= A + B + C, Z Z Φ A= dv∗ g∗ dvf τ−v∗ ◦ T∆ ◦ τv∗ {pk ψj2 pk f }, Rn Rn Z Z B= dv∗ g∗ dvf τ−v∗ ◦ T 1 ◦ {[Φ, pk ]τv∗ ψj2 pk f }, Rn

Rn

Z

Z

C=

dv∗ g∗

Rn

dvf τ−v∗ ◦ T 1 (pk Φτv∗ ψj2 pk f ).

(93) (94)

(95)

Rn

Φ Above, T∆ is defined as in [8, 12] by Z v Φ T∆ φ(v) = φ(v + )[Φ(v + ) − Φ(v)]b( .σ)dσ. |v| n−1 S

By using computations on the Fourier side, one can show that C = D + E,

(96)

36


where Z Z Z D= dv∗ g∗ dξ

n o n \ o + ξ .σ) ψk (ξ) − ψk (ξ + ) e−iv∗ .ξ fˆ(ξ + ) Φτv∗ ψj2 pk f , |ξ| Rn S n−1 (97) Z Z E= dv∗ g∗ dv pk f τ−v∗ ◦ T 1 ◦ (Φτv∗ ψj2 pk f ). (98)

Rn

and

dσb(

Rn

Rn

We then write where

F =− and G=

Z

Z

E = F + G,

Z Φ τv∗ ψj2 pk f, dv∗ g∗ dvpk f τ−v∗ ◦ T∆

Rn

(100)

Rn

Z

dv∗ g∗

Rn

(99)

dvpk f τ−v∗ ◦ T Φ ◦ τv∗ ψj2 pk f,

Rn

the last term being also given as Z Z Z v − v∗ G= dv∗ dv dσg∗ pk f Φ(v − v∗ )b( .σ)[(ψj2 pk f )′ − (ψj2 pk f )]. |v − v∗ | Rn Rn S n−1 For this last term, one has G = H + I, where Z Z Z v − v∗ H= dv∗ g∗ dv dσb( .σ)pk f Φ(v − v∗ )(ψj′ − ψj )(ψj pk f )′ , |v − v | n n n−1 ∗ R R S and Z Z Z v − v∗ I= dv∗ g∗ dv dσb( .σ)ψj pk f Φ(v − v∗ )[(ψj pk f )′ − (ψj pk f )]. |v − v∗ | n n n−1 R R S

(101) (102) (103)

(104)

By using the simple identity a(b − a) = − 12 (b − a)2 + 12 (b2 − a2 ), it follows that I = J − K,

where J =

1 2

and

Z

Z Z dv∗ g∗ dv

Rn

Rn

dσb( S n−1

(105)

v − v∗ .σ)Φ(v − v∗ )[((ψj pk f )′ )2 − (ψj pk f )2 ], (106) |v − v∗ |

Z Z Z 1 v − v∗ dv∗ g∗ dv dσb( .σ)Φ(v − v∗ )[(ψj pk f )′ − (ψj pk f )]2 . (107) 2 Rn |v − v∗ | Rn S n−1 Thus, we have obtained the following differential equality d kψj pk f k2L2 + K = A + B + D + F + H + J . (108) dt It remains to find upper bounds on each term on the right hand side, and a lower bound on K. By using the Cancellation Lemma 4.1, since we have assumed all moments on f (and thus on g) bounded, we find K=

|J | . 2j kψj pk f k2L2 . (109) Using Taylor’s formula, and denoting by ˜b = b sin θ2 , we note that ˜b is integrable over the unit sphere, and so we are in a situation similar to the usual cutoff gain term, so that, see for instance [8, 112]

BOLTZMANN EQUATION

|H| .

1 2j(γ+2)

kgkL1γ+1 kpk f kL2 kψj pk f kL2 .

37

(110)

As in [8], one can show that |F | . kgkL1γ kpk f kL2γ kψj pk f kL2 .

(111)

|A| . kgkL1γ kf kL2γ kψj pk f kL2 ,

(112)

|B| . kgkL1 kf kL2 kψj pk f kL2 .

(113)

|D| . kgkL1γ k˜ pk f kL2 2jγ kψj pk f kL2 .

(114)

In the same way,

and similarly

By using another Littlewood-Paley partition, and making the change of variables ξ+ → 7 ξ as in [9], we get For the lower bound on K, from Peetre’s inequality (this is where the smoothed version of the kinetic part is needed), it follows that, using similar computations as those from [9], that is the coercivity estimate given by Theorem 5.1 and a commutator estimate, K & 2jγ 2kν kψj pk f k2L2 − C2j(γ−2) 2k(ν−2) kpk f k2L2α ,

(115)

for all α ≥ 0. Collecting all the above estimates, we have found that, setting Uj,k = kψj pk f k2L2 , one has  ∂Uj,k + C2jγ 2kν Uj,k .   1 1  j  2 2 2 Uj,k + 2−j(γ+2) kg|L1 kpk f kL2 Uj,k + kgkL1γ kpk f kL2γ Uj,k γ+1 1 1 (116) 2 2  +kgkL1γ kf kL2γ Uj,k + kgkL1 kf kL2 Uj,k    1  2 +kgkL1γ k˜ pk f kL2 2jγ Uj,k + 2j(γ−2) 2k(ν−2) kpk f k2L2α . ν

2 Then, in a first step, one can show that f ∈ B2,2,α , for all t ≥ t1 > t0 . In a second step, to improve this small regularity, we need to go back to the previous different terms, using this small regularity, to get improved upper bounds. Then ν one can show that f ∈ B2,2,α for all α ≥ 0, by the same type of arguments, and so the result by iteration. This concludes the proof.

Remarks 8.2. 1. The assumption ν ∈ (0, 1) was only made for convenience, since we have used results from [8]. The range ν ∈ [1, 2) is in fact avalaible, see [12] and the results recalled in Section 7. We have also considered a smoothed version of the kinetic kernel. It should be certainly possible to consider truly the real case |v|γ , by using in particular technics from [61]. 2. Next, what about relaxing assumption (91)? First of all, note that L2α space therein can be replaced by any Lpα space, but for a p > 1. This fact comes from the proof, in particular using Holder instead of Cauchy-Scharwz inequality, and then Bernstein’s inequality to recover L2 norms. This is at this point that the choice p = 1 does not (seem to) work, because of the bad exponent coming from Bernstein’s inequality, for which iteration fails to improve the regularity.

38


Note also that Assumption (91) can be certainly relaxed as regards number of moments. That is, only a finite number of moments is really needed to initiate the iteration process. Next, adding the assumption of boundedness on entropy dissipation rate (which is in fact part of the definition of an entropic weak solution), we can assume that Z T γ p k < v > 2 f (s)k2H ν2 ds < +∞. (117) 0

From the books quoted in the bibliography, in particular [116], we get Z T k < v >γ f (s)kHps1 ds < +∞, (118) 0

n where p1 = > 1, but p1 < 2 . n − ν2 For example, let us forget about integrability w.r.t. time t. Then it follows n from Sobolev embedding that < v >γ f ∈ Lp2 , where p2 = n−ν . Of course n p2 > 1, but we note that p2 ≥ 2 iff ν ≥ 2 . In particular, in dimension n = 2, this is the case iff ν ≥ 1, while in dimension 3, this is the case iff ν ≥ 32 . In conclusion when ν is really very close to 2, then this L2 bound is available. All in all, in dimension n = 2 or n = 3, it should be certainly possible (with some extra work concerning the integrability w.r.t. time) to relax assumption (91) and get our result. We also note, that having in mind [52, 51], small power of f should have good regularity. These remarks explain also the fact that Landau equation, corresponding to a version of Boltzmann equation with ν = 2 is somehow easier to deal with, see for instance [54, 61], starting from an entropic solution, and thus without any further assumption. 8.2. Further improvements. The above results were assuming a control of moments in L2 . This assumption was removed by Huo and coll. [87]. Here are the main results from that paper, and we refer also to [87] for the extensions therein to the Debye Ukawa potential Theorem 8.3. Assume that 0 ≤ γ ≤ 1 and 0 < ν < 1. Let f be any weak solution of the spatially homogeneous Boltzmann equation, with the usual entropic bounds, and the mass conservation. Then f ∈ L∞ ([t0 , T ], H +∞ (R3 ))

for any t0 > 0, T > 0. Further, if we assume that

|v|m f ∈ L∞ ([T0 , T1 ], L1 (R3 ))

for all m ∈ N, for all 0 ≤ T0 < T1 (which means propagation of moments in the case T0 = 0 and gain of moments in the case T0 > 0, then for all t0 > T0 .

f ∈ L∞ (]t0 , T1 ]; S(R3 )),

The proof of this result uses: 1) the functional estimates mentioned in Section 7 on the Boltzmann collision operator and related collision operators; 2) commutators, multipliers and pseudo-differential technics.

BOLTZMANN EQUATION

39

The second point avoids in particular completely the use of the truncation in velocity variables used in the previous sub-section, but is therefore technically much more involved. We refer to [87] for complete proofs, but we shall sketch some of the main ideas later on when we will consider the spatially inhomogeneous Boltzmann equation later on. 9. Spatially homogeneous Landau equation: Regularity. We shall now consider the Landau equation, which is closely related to the Boltzmann equation. Understanding more precisely the Landau case is clearly helpful for getting clues about the Boltzmann case. For example, we shall see that we can deal with real hard cross-sections, without assuming a smoothing property, as mentioned previously in the Boltzmann case. Recent works have improved our knowledge about this equation: see Desvillettes and Villani [54] for the spatially homogeneous case, [76] for the global existence of solutions close to global equilibrium in the spatially inhomogeneous case, see also [110], and see also in the context of renormalised solutions the works [128, 17]. The spatially inhomogeneous Landau equation was also considered recently by Desvillettes and coll [39], that we shall comment later on. In this Section, we shall concentrate on regularization issues in the spatially homogeneous case. Other questions will be reviewed later on. Regularization questions have been already studied as a particular question in the spatially homogeneous case [54], but we shall revisit this problem under the same framework, that is any weak solution is indeed smooth. The spatially homogeneous Landau equation writes as ∂f (v, t) = Q(f, f )(v, t), (119) ∂t with a given non negative initial data f (0, v) = f0 (v), and Q(f, f )(v, t) = Z n ∂f ∂f 1 X ∂ { dv⋆ aij (v − v⋆ )[ f (v⋆ , t) (v, t) − f (v, t) (v⋆ , t) ]}, 2 i,j=1 ∂vi Rn ∂vj ∂v⋆j

(120)

with

(aij (z))1≤i,j≤N = Λ | z |2 χ(| z |)Πij (z). (121) We have used implicit summation convention over repeated indices (i, j). Above, Λ is some positive constant that we normalize to be 1, χ is a non negative function, and Π(z) is the following non negative symmetric matrix, corresponding to the orthogonal projection onto z ⊥ zi zj Πij (z) = δij − 2 . |z|

Landau collision operator (120) has properties similar to Boltzmann operator, in particular preserving mass, momentum and energy, and also satisfying the well known Boltzmann’s H-theorem. We assume that function χ, entering Landau operator, is smooth and satisfies γ

β

γ

c(1 + |z|2 ) 2 ≤ χ(|z|) ≤ C0 (1 + |z|2 ) 2 , 2

|∇ (χ(|z|))| ≤ Cβ (1 + |z| )

γ 2

−β

for all multi-index β,

(122) (123)

40


where c and Cβ are positive constants, latter one depending on β. Furthermore, we restrict herein to so-called hard potentials, i.e. 0 < γ ≤ 1. However, note that this assumption could be relaxed to γ ≤ 2, and even to any γ > 0, if we make some minor changes on initial data. We shall also consider so-called very hard potentials, that is χ(| z |) =| z |γ for 0 < γ ≤ 1, (124) corresponding to interaction laws with inverse s-power forces for s > 2n−1. It differs slightly but importantly from (123), due to the singularity around zero. From now on, we assume that a weak solution to Landau equation (119) has been constructed, satisfying usual entropic estimate, for a fixed T > 0 (eventually T = +∞) Z sup f (t, v)(1 + |v|2 + log(1 + f (t, v))dv < ∞. (125) t∈[0,T ]

Rn

According to known propagation of moments results [54], in order to simplify the exposition, we shall assume these moments estimates on initial data, and thus on solutions themselves: For all s > 0, || f0 ||L1s < +∞, and sup || f (t) ||L1s < +∞.

(126)

t≥0

It is in fact possible to avoid this assumption on initial data and use instead known immediate propagation of moments, at the expense of an adaptation of the proofs of [61]. We shall show an S regularity for any weak solution. Thus, we do not need uniqueness result to conclude: any weak solution is smooth. Again, the proof uses only elementary arguments. In particular, Sobolev interpolation inequalities are avoided, using instead Littlewood-Paley harmonic decomposition. Before stating the main result, let us firstly point out basic notations used herein. We define, for z ∈ Rn bi (z) = ∂j aij (z) i.e. b(z) = ∇a(z)

(127)

2

c(z) = ∂ij aij (z) i.e. c(z) = ∇ a(z), and adopt the following notations a = a ∗ f, b = b ∗ f and c = c ∗ f.

We can then write (119) alternatively under the form ∂t f = aij ∂ij f − cf

i.e. ∂t f = a∇2 f − cf.

(128)

L1s

For all s ≥ 0, we use weighted spaces and their standard norms Z s || f ||L1s ≡ | f (v) | (1+ | v |2 ) 2 dv. Rn

Moreover, we set

M0 =

Z

Z 1 f0 (v)dv, E0 = f0 (v) | v |2 dv, 2 Rn Rn Z H0 = f0 (v) log f0 (v)dv, Rn

for initial mass, energy and entropy of initial data. The usual entropic space is defined by Z | f (v) || log(| f (v) |)dv < +∞}. L log L(Rn ) = { f ∈ L1 (Rn ); Rn

BOLTZMANN EQUATION

41

Commutator of two operators A and B is defined as [A, B] = AB − BA. Then, the main result from [61] is given by Theorem 9.1. Let f0 ∈ L12+δ ∩ LlogL(Rn ), for some δ > 0. Let f be any weak non negative solution of Landau spatially homogeneous equation (119), (120) satisfying assumptions (122) or (124) and (126). Then, f lies in C ∞ ([t0 , +∞]; S(Rn )) for all t0 > 0. Proof. We only sketch the main lines, under assumption (123). Recall Landau equation is given by (128) ∂t f = a ¯∇2 f − cf.

(129)

We let k and j be any positive integers. We agree all functions ψj appearing with negative indices, are null functions. Hereafter, for simplicity, variables m, m′ will notify a finite summation on m and ′ m . More precisely, variable m (resp. m′ ) denotes summation on m ( resp. m′ ) from j − 2 to j + 2. A simple computation using Littlewood Paley, and integrating by parts, yield the (long!) equality Z 1 aψ˜j ∇2 pk (ψj f )pk (ψj f ) (130) ∂t || pk (ψj f ) ||2L2 − 2 Z 1 = [ ∇2 pk , aψ˜j ](ψj f )pk (ψj f ) 2 Z Z − [ ∇pk , ∇(aψ˜j )](ψj f )pk (ψj f ) − ∇(aψ˜j )∇pk (ψj f )pk (ψj f ) Z Z 1 1 + [ pk , a∇2 ψ˜j + 2b∇ψ˜j ](ψj f )pk (ψj f ) + (a∇2 ψ˜j + 2b∇ψ˜j )pk (ψj f )pk (ψj f ) 2 2 Z Z − [ ∇pk , a∇ψj ](ψm f )pk (ψj f ) − a∇ψj ∇pk (ψm f )pk (ψj f ) Z 1 + [ pk , a∇2 ψj + 2b∇ψj ](ψm f )pk (ψj f ) 2 Z 1 + (a∇2 ψj + 2b∇ψj )pk (ψm f )pk (ψj f ) 2 Z 1 + [ pk , a∇ψj ∇ψm ](ψm′ f )pk (ψj f ) 2 Z 1 + a∇ψj ∇ψm pk (ψm′ f )pk (ψj f ). 2 Firstly, integrating by parts , second term on first line of (130) becomes Z 1 (aψ˜j )∇2 pk (ψj f )pk (ψj f )dv = − 2 v (131) Z Z 1 1 ∇(aψ˜j )∇pk (ψj f )pk (ψj f )dv + (aψ˜j )∇pk (ψj f )∇pk (ψj f )dv. 2 v 2 v It is then possible to give a lower bound on the second term on the r.h.s. on (131). In particular, it follows that γ

aij (v)ξω ξj ≥ K(1+ | v |2 ) 2 | ξ |2

∀ξ ∈ Rn ,

where K is a non negative constant depending only on the initial moment M0 , energy E0 , entropy H0 and γ. Taking into account support of our basic functions,

42


we obtain γ aψ˜j ∇pk (ψj f )∇pk (ψj f ) ≥ KCj2 ψ˜j ∇pk (ψj f )∇pk (ψj f ) where Cj = (1 + 22j ). Since

ψ˜j ∇pk ψj f = ∇(pk ψj f ) − [∇pk , ψ˜j ]ψj f, it follows that, integrating by parts Z γ 1 aψ˜j ∇pk (ψj f )∇pk (ψj f )dv ≥ CCj2 ||∇pk (ψj f )||2L2 2 Z Z γ γ 2 +CCj2 [ ∇ pk , ψ˜j ]ψj f pk (ψj f )dv − CCj2 (∇ψ˜j )∇pk (ψj f )pk (ψj f )dv.

(132)

The other terms are then upper bounded as follows. Note that pd ˜k g is supported in 2k−2 ≤ |ξ| ≤ 2k+2 , for any function g ∈ L1 , so that k+3 X

||˜ pk g||L2 ≤

l=k−3

||˜ pk (pl g)||L2 ≤ C

k+3 X

l=k−3

||pl g||L2 .

(133)

Next, taking into account the smoothness assumption on a, using the precise support sets of ψj , we obtain, for all j ∈ N γ+2 2

| aψj |≤ CCj

γ+1 2

, | a∇ψj |≤ CCj γ+1 2

| bψj |≤ CCj

γ

, | a∇2 ψj |≤ CCj2 ,

(134)

γ

, | b∇ψj |≤ CCj2 γ

and | ∇β (aψj ) |≤ CCβ Cj2 , ∀β ≥ 2. Inequalities (134) hold also true if we change ψj by ψ˜j . Integrands in (130) and (132) are of the form [ ∇q pk , φ], for order q = 0, 1, 2, where φ is any smooth function, and they can be bounded by using simple commutator type estimations. Then, together with Bernstein’s inequality, one gets, for all integers k ≥ 0, j ≥ 0, for a fixed integer r > 0 γ+2 2

r ∂t Ukj (t) + Ckj Ukj (t) ≤ Kkj (t) + CCj

where Ukj =

{ Ukj (t) +

k+3 X

l=k−3

j+2 X

h=j−2,h6=j

or

l6=k

Ulh (t) }, (135)

γ ||pk (ψj f )||2L2 2 2k , C = CC 2 , Cj = (1 + 22j ) kj j 2k(n+1) γ

Cj2 and = C rk || pk (ψj f )(t) ||L1 . 2 It follows that, for any integer r > 0, for all t > t0 > 0 ∞ X ∞ X β Ckj Ukj (t) < Ct0 , r Kkj (t)

k=0 j=0

where Ct0 is a constant depending only on || f ||L1γ and || f ||2L1 . And iterating, γ+2 2 P∞ P∞ qβ one can show that the series k=0 j=0 Ckj Ukj (t) is finite, for all integer q, as γ h soon as t > t0 , that is f ∈ Hs for all h > 0, s ≥ 2β (h + n + 1). All in all, we see h that (still when t > 0) f lies Hs for all h, s > 0 and therefore lies in S.

BOLTZMANN EQUATION

43

For the proof of Theorem 9.1, under assumption (124), that is for the hard potentiel case χ(| v |) =| v |γ , which has a singularity around zero, the method of proof used above still applies for this case, up to some modifications that we now explain. The main troubles are due to the commutators estimates. In order to get ride of the singularity near 0, we split the kernel into two parts. First one will be smooth, thus avoiding the singularity and therefore we can apply on it the manipulations of the iteration’s method. For the second one, it is possible to make it as small as one wishes, in a suitable sense. We refer to [61] for more details about the exact proof. It is certainly possible to apply the methods from [87], but the above proof needs really only basic knowledge from standard Analysis tools. 10. Back to coercivity: Linear or linearized case. For reasons which will be made clear later on, it is important to understand the case of the linear or linearized Boltzmann operator, for a linearization close to a Maxwellian. At least for the study of regularization properties, it is also important to improve the results of [9], as recalled in Section 5. 10.1. The linear Boltzmann case (ω representation). Let us first begin with the linear case (extracting some results from [4]), in dimension 3. We are now using ω−representation. In view of Section 3 on the pseudo-differential formulation and of Section 4 related to the cancellation Lemma, note first of all that Theorem 10.1. Let L be the following Boltzmann 3D linear operator L(f )(v) ≡

Z

′ 2

2 R3v1 ×Sω

2

dv1 dω{f (v ′ )e−|v1 | − f (v)e−|v1 | }B(| v − v1 |, | (

v − v1 , ω) |). | v − v1 |

Then, one has the decomposition L(f )(v) = L1 (f )(v) + L2 (f )(v) + Lc (f )(v), where L1 (f )(v) =

−Cs′

Z

2

R3α

L2 (f )(v) = +Cs′ .f (v). Lc (f )(v) =

Z

dαe−|α+v| | α |γ+ν | S(α).D |ν−1 (f )(v),

R3v1

Z

Z

R3α

2 Sw

2

dα | α |γ+ν | S(α).D |ν−1 (e−|.| )(α + v), ′ 2

2

dv1 dw{f ′ e−|v1 | − f e−|v1 | }B c (., .),

with the same notations and assumptions as in Section 3. ω-representation is used above. Observe that the operator Lc corresponds exactly to a cutoff type Boltzmann operator. Furthermore, operator L2 can be analyzed through the use of the Cancellation Lemma, see Section 4. In view of these facts, both Lc and L2 do not lead to any regularity improvement. Therefore, it is important to understand the structure of the operator L1 . By the way, let us also note the following extensions of Theorem 10.1, when assuming B satisfies

44


v − v1 1 , ω) |) ≡| v − v1 |γ .χ(| v1 − v ′ |), v−v1 | v − v1 | | ( |v−v1 | , ω) |ν

Bχ (| v − v1 |, | (

(136)

where χ is a smooth given function: the decomposition stated in Theorem 10.1 above holds true, provided we change the weight | α |γ+ν into | α |γ+ν χ(| α |) and 1 v−v1 Bc into Bc χ(| v1 − v | {1− | ( |v−v , ω) |2 } 2 ). 1| Such general χ does not influence on the non angular cutoff hypothesis made on B. We first set the following Definition 10.2. For v ∈ R3 , ξ ∈ R3 , let Z 2 a(v, ξ) ≡ Cs′ dαe−|α+v| | α |γ+ν | S(α).ξ |ν−1 = R3 Z α 2 = Cs′ dαe−|α+v| | α |γ+1 | α ∧ ξ |ν−1 R3α

so that (pdo notations)

L1 (f )(v) = −a(v, Dv )(f )(v),

and finally

a ˜(v, ξ) = a(v,

ξ ), f or ξ ∈ S 2 . |ξ|

The following Lemma gives suitable estimates on a,

Lemma 10.3. The symbol a(v, ξ) given by Definition 10.2 satisfies C1 (1+ | v |2 )

γ+1 2

(| ξ |2 + | ξ ∧ v |2 ) 2

≤ C2 (1+ | v | ) if s ≥ 3 (γ + 1 ≥ 0) and C1′ (| ξ |2 + | ξ ∧ v |2 )

γ+ν 2

if s ≤ 3 (−2 < γ + 1 ≤ 0).

|ξ|

−γ−1 2

γ+1 2

2

ν−1 2

≤ a(v, ξ) ≤

(| ξ | + | ξ ∧ v |2 )

ν−1 2

,

≤ a(v, ξ) ≤ C1′ (| ξ |2 + | ξ ∧ v |2 )

γ+ν 2

|ξ|

−γ−1 2

,

Proof. From Definition 10.2, one has a(v, ξ) =

Cs′

Z

R3α

2

dαe−|α| | α − v |γ+1 | S(ξ).α − S(ξ).v |ν−1 | ξ |ν−1 .

(137)

Now recall ν − 1 > 0 and γ + 1 = 2(s−3) s−1 , so that γ + 1 ≥ 0 iff s ≥ 3. Therefore, for s ≥ 3, one has, using Holder inequality γ+1

ν−1

a(v, ξ) ≤ C(1+ | v |2 ) 2 (| ξ |2 + | ξ ∧ v |2 ) 2 . (138) Now, still for s ≥ 3, we turn to the lower bound. We start from (137) and writing ξ α and v in a fixed orthonormal basis of R3 with |ξ| as first vector, we obtain Z 2 ν−1 ′ a(v, ξ) = Cs e−|α| | α − v |γ+1 | (α − v)22 + (α − v)23 | 2 | ξ |ν−1 R3α

BOLTZMANN EQUATION

Z

≥ C | ξ |ν−1

45

2

R3α ,|(α−v)22 +(α−v)23 |≥1

Z

ν−1

≥C|ξ|

e−|α| | α − v |γ+1 | (α − v)22 + (α − v)23 |

2

.

e−|α| (1+ | α − v |2 )

γ+1 2

| 1 + (α − v)22 + (α − v)23 |

ν−1 2

ν−1 2

.

Using Peetree’s inequality, it follows that for, say | v | ≥ 2, one has Z 2 1 1 ν−1 a(v, ξ) ≥ C | ξ | e−|α| ν−1 × γ+1 2 2 (1+ | α | ) 2 (1 + α2 + α23 ) 2 . ×(1+ | v |2 )

γ+1 2

| 1 + (v22 + v32 ) |

ν−1 2

.

Therefore, we get an uniform estimate for big | v |, and as a(v, ξ) does not take value 0 for small | v |, it follows a(v, ξ) ≥ C(1+ | v |2 )

γ+1 2

(| ξ |2 + | ξ ∧ v |2 )

ν−1 2

,

(139)

which is the first part of the Lemma. For 2 < s ≤ 3 (−2 < γ + 1 ≤ 0), one has a(v, ξ) = C

Z

2

.

e−|α+v| {( ≤C 2

≤ C(1+ | S(ξ).v | )

γ+ν 2

Z

.

γ+1 ξ .α)2 + | S(ξ).α |2 } 2 | S(ξ).α |ν−1 | ξ |ν−1 |ξ| 2

e−|α+v| | S(ξ).α |γ+ν | ξ |ν−1

| ξ |ν−1 ≤ C(| ξ |2 + | ξ ∧ v |2 )

γ+ν 2

| ξ |−

γ+ν 2

.

(140)

For the lower bound, one can proceed as follows Z 2 ′ a(v, ξ) = Cs e−|α| | α − v |γ+1 | S(ξ).(α − v) |ν−1 | ξ |ν−1 . Z 2 dαe−|α| | α − v |γ+1 | S(ξ).(α − v) |ν−1 | ξ |ν−1 ≥ Cs′ R3α ,|S(ξ).(α−v)|≥1

≥C

Z

.

2

e−|α| (1+ | α − v |2 )

γ+1 2

| S(ξ).(α − v) |ν−1 | ξ |ν−1 ,

and for big | v |, and again by Peetree’s inequality, we obtain a(v, ξ) ≥ C | ξ |−

γ+1 2

(| ξ |2 + | ξ ∧ v |2 )

γ+ν 2

.

(141)

Note that the Maxwellian could be changed by any other function in S (in fact L1 with some weight is enough), and therefore proceeding exactly as above, we obtain Lemma 10.4. The symbol a(v, ξ) given by Definition 10.2 satisfies | ∂vm a(v, ξ) | ≤ Cm (1+ | v |2 )

if s ≥ 3 (γ + 1 ≥ 0), and

γ+1 2

(| ξ |2 + | ξ ∧ v |2 )

| ∂vm a(v, ξ) | ≤ Cm (| ξ |2 + | ξ ∧ v |2 )

if 2 < s ≤ 3 (−2 < γ + 1 ≤ 0).

γ+ν 2

| ξ |−

ν−1 2

γ+1 2

,

,

Let us turn to a, and analyse its smoothness with respect to variable ξ, with the aim at getting functional properties of the operator a ˜(v, Dv ).

46


Lemma 10.5. For 2 < s < 3, one has

and for any s > 2,

k ∂vm a ˜(v, .) kW 2,2 (S 2 ) ≤ Cm (1+ | v |2 )

γ+ν 2

k ∂vm a ˜(v, .) kW 2,1 (S 2 ) ≤ Cm (1+ | v |2 )

γ+ν 2

,

.

Proof. We only consider the case m = 0. Letting Λ for the Laplace operator on S 2 , one has, by Minkowski’s inequality, Z 1 { | Λ˜ a(v, ξ) |2 dξ} 2 S2 Z Z 2 ξ ν−1 2 1/2 =( dξ | dαe−|α+v| | α |γ+ν Λ{| S(α). | }| ) | ξ | 2 3 S Rα Z Z 2 ξ ν−1 2 1/2 ≤ dα( dξe−2|α+v| | α |2(γ+ν) | Λ{| S(α). | }| ) |ξ| R3α S2 Z Z 2 ξ ν−1 2 1/2 ≤ dαe−|α+v| | α |γ+ν { dξ | Λ{| S(α). | }| } . (142) | ξ | 3 2 Rα S

Clearly, the last integral over S 2 does not depend on α, so that shifting to polar coordinates, with first axis directed by α, with angle (θ, φ), 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π, ξ ν−1 we note that | S(α). |ξ| | reads u(θ) =| sinθ |ν−1 . For functions depending only on θ, the operator Λ acts as Λu = (sinθ)−1 ∂θ (sinθ∂θ u), and it follows Λu = (ν − 1)2 (sinθ)ν−3 cos2 θ − (ν − 1)(sinθ)ν−1 ≡ A + B. Now, it is easy to check that Z Z π 2 | B | dω ≤ C (sinθ)2ν−1 dθ < +∞, S2

as 2ν − 1 =

s+3 s−1

0

> 0. As for A, one has Z Z π | A |2 dω ≤ C (sinθ)2ν−5 dθ < +∞, S2

0

if and only if 2 < s < 3. Indeed, one finds that 2ν − 5 = −3s+7 s−1 > −1, in this case. Note this value also appears in [72] (for different reasons). For the k . kW 2,1 bounds, it is enough to note that Z Z π | A | dω ≤ C (sinθ)ν−2 dθ < +∞, S2

as ν − 2 =

−s+3 s−1

0

> −1, and Z Z π | B | dω ≤ C (sinθ)ν dθ < +∞. S2

0

The above results explains the difficult behavior of this symbol, which does not belong to a classical Hormander’s class. However, the most important result is given by Lemma 10.3 which gives exactly similar lower and upper bounds. It explains why, at least in the linear or linearized case, the lower bound given by the Coercivity result, starting from the entropy dissipation estimate, needs to be improved.

BOLTZMANN EQUATION

47

10.2. The linearized Boltzmann and Landau case. With this last comment in mind, we now turn to the linearized Boltzmann and Landau operator, and to the results by Mouhot and coll. [23, 108, 111]. Though these results improve and give explicit lower bounds for the linearized Boltzmann operator in the cutoff case, the singular case is still open, in view of the results displayed in the previous sub-section. We go back to the usual σ−representation, and we start by the results of Baranger and Mouhot [23]. 2 Letting f be a perturbation of the global Maxwellian M = e−|v| , f = M (1 + h), the associated Boltzmann linearized collision operator is given by LB0 h(v) =

Z

Rn

Z

S n−1

B(cos θ, |v − v∗ |)M (v∗ )[h′∗ + h′ − h∗ − h]dσdv∗ .

(143)

The corresponding entropy is then the L2M (weighted by M ) norm of h and thus the entropy dissipation functional is given by DB0 (h) = − < h, LB0 h >L2M

Z Z Z 1 B(cos θ, |v − v∗ |)[h′∗ + h′ − h∗ − h]2 M M∗ (144) 4 Rn Rn S n−1 which is thus positive. In particular, it follows that the spectrum of LB0 in L2M is nonpositive. Similarly, the Landau collision operator is given by Z QLa (f, f )(v) = ∇v .( dv∗ a(v − v∗ )[f∗ (∇f ) − f (∇f )∗ ]dv∗ , (145) =

Rn

2

where a(z) = |z| Φ(z)Πz⊥ , Πz⊥ being the orthogonal projection onto z ⊥ that is zi zj (Πz⊥ )i,j = δi,j − 2 . |z| Then the corresponding linearized Landau operator is Z La −1 L h(v) = M (v) ∇v .( a(v − v∗ )[(∇h) − (∇h)∗ ]M M∗ )

(146)

Rn

and the linearized Landau entropy dissipation functional is given by

=

1 2

RR

DLa (h) = − < h, LLa h >L2M

(147)

Φ(v − v∗ )|v − v∗ |2 ||Π(v−v∗ )⊥ [(∇h) − (∇h)∗ ]||2 M M∗ dv∗ dv

which is also positive, and thus the spectrum of LLa in L2 (M ) is also nonpositive. The assumptions made in [23] on the collision kernel B are as follows. First of all, B is given by B = b(cos θ)Φ(|v − v∗ |) (148) for nonnegative functions Φ and b. The kinetic part Φ is then assumed to be bounded from below at infinity (so that it works for maxwellian or hard potentials): for some R ≥ 0, cφ > 0, then Φ(r) ≥ cΦ for all r ≥ R and the angular part b satisfies Z cb = inf min{b(σ1 .σ3 ), b(σ2 .σ3 )}dσ3 > 0, σ1 ,σ2 ∈S n−1

σ3 ∈S n−1

(149)

(150)

48


which does cover all physical cases of interest. Previous studies of spectral gaps were mainly based on Grad’s cutoff assumption together with an application of Weyl’s theorem for the linearized operator, written as a compact perturbation of a multiplicative operator. However, Pao [113] has already considered the non cut off case hard potentials. Let us note that for soft potentials, Boltzmann operator has no spectral gap, but does have if we change slightly the lower norm (a weighted space), see [23] for further details. The method of Baranger and Mouhot is explicit, and applies to hard potential (and maxwellian) with or without cutoff. The important point of this paper is to reduce by a very clever argument (originally due to Villani) to the Maxwellian case. We just consider the Boltzmann case, referring to [23, 108] for the Landau case: Theorem 10.6. Under assumptions (148), (149) and (150), the Boltzmann entropy dissipation fucntional DB0 satisfies for all h ∈ L2M 2

DB0 (h) ≥ (

cΦ cb e−4R )D0B0 (h) 32|S n−1

(151)

where D0B0 (h) stands for the case B = 1. It follows that one has the following spectral gap estimate: for all h ∈ L2M orthogonal in that space to 1, v and |v|2 , DB0 (h) ≥ C||h||2L2

M

(152)

where C is an explicit constant.

One important step of the proof is to reduce the case of hard potentials to the maxwellian case. Then, one tries to get the following inequality:  RR |ξ(x) − ξ(y)|2 |x − y|γ M (x)M (y)dxdy  (153) RR  ≥ Kγ |ξ(x) − ξ(y)|2 M (x)M (y)dxdy for γ ≥ 0, some function ξ and where Z Z 1 R Kγ = min{|x − z|γ , |z − y|γ }M (z)dz. 4 M x,y As mentionned in [23], the proof of such an inequality uses the existence of a function F = F (x, y) which satisfies a triangle inequality. Above F = |ξ(x) − ξ(y)|2 which satisfies F (x, y) ≤ 2F (x, z) + 2F (z, y)

Therefore, another main point is to get such a triangular inequality, and this is made by a clever introduction of intermediate collisions, or mathematically speaking: shift to polar coordinates and then introduce a nice “blind” spherical variable, that allows first to remove the angular part b, and then to remove the kinetic part Φ, which by assumption, could be degenerate for small relative velocities. These results were then extended by Mouhot [108], with the following assumptions, in dimension n: B = Φ(v − v∗ )b(cos θ)

(154)

BOLTZMANN EQUATION

49

for nonnegative functions Φ ans b. The kinetic part is assumed to be bounded below by a power law Φ(r) ≥ CΦ rγ for some γ ∈ (−n, 1], while the angular part b satisfies Z Cb = inf n−1 min{b(σ1 .σ3 ), b(σ2 .σ3 )}dσ3 > 0 σ1 ,σ2 ∈S

(155)

(156)

S n−1

and in the case of non integrable kernels,

∀θ ∈ (0, π], b(cos θ) ≥

cb n−1+ν θ

(157)

for some constant cb > 0 and ν ∈ [0, 2) The results of Mouhot [108] are then given by Theorem 10.7. Under assumptions (154), (155), (156), one has γ

∀h ∈ L2M , DB (h) ≥ C||h − Π(h)|| < v > 2 ||2L2

(158)

M

for an explicit constant C. Moreover, under assumptions (154), (155) and (157), one has γ

∀h ∈ L2M , DB (h) ≥ C||h − Π(h)|| < v > 2 ||2H α/2

(159)

loc

Above Π is the orthogonal projection of the null space of LB in L2M , that is N (LB ) = Span{1, v, |v|2 } Note that the last result is not a global estimate, and so it is still open to get a global weighted estimate, see below in the Landau case. Proof of this result uses previous estimates from [23], together with suitable truncation and change of variables, as now usual from the ideas of [9], see Section 5. In the Landau case, the following result was proven by Mouhot [108], but is originally due to Guo [76] Theorem 10.8. (linearized Landau operator). Under assumptions (155), one has ∀h ∈ L2M , DL (h) ≥ C(||[h − Π(h)]||2H 1

γ M

+ ||[h − Π(h)] < v >1+γ/2 ||2L2

M

(160)

where C is an explicit constant. In fact, a further step was settled down by Mouhot and Strain [111]. This time, we make the following assumptions:

with

B = |v − v∗ |γ b(cos θ), γ ∈ (−n, +∞)

(161)

b(cos θ) ∼θ∼0 b∗ (θ)(sin θ/2)−(n−1)−ν , ν ∈ [0, 2) (162) and assume w.l.o.g. that it is supported on the set θ ∈ (π/2, π). It is convenient to change slightly the linearization. Introduce the maxwellian 2 µ = µ(v) = (2π)−n/2 e−|v| /2 which is thus the unique normalized equilibrium with mass 1, momentum 0 and temperature 1. Then we consider linearization of f as f = µ + µ1/2 g for which the corresponding linearized operator writes

50


LB (g) = −µ−1/2 [QB (µ, µ1/2 g) + QB (µ1/2 g, µ)] where QB still denotes the Boltzmann collision operator with kernel B. This slight change has the advantage that we can work directly with L2 space, contrary to the weighted space L2M used previously. In this case, LB is an unbounded symmetric operator on L2 (w.r.t. velocity variables), such that DB (g) ≡ (LB g, g) ≥ 0 and DB (g) = 0 iff g = Pg, where Pg denotes the orthogonal projection (in L2 ) on the space of collisional invariants n o Span 1, v, |v|2 µ1/2 . The same linearization will be used also for the linearized Landau operator. Here is the coercivity result proven in the case of Boltzmann operator [111]

Theorem 10.9. Let B be a collision kernel satisfying (161) and (162). Then 1) for any ε > 0, there exist an explicit constant, such that DB (g) ≥ C||[g − Pg](1 + |v|2 )(γ+ν−ε)/4 ||2L2 . 2) There exists a non constructive constant C such that DB (g) ≥ C||[g − Pg](1 + |v|2 )(γ+ν)/4 ||2L2 . In the Landau case, one has [111] Theorem 10.10. Let γ ∈ [−n, +∞). Then, there exists an explicit (constructive) constant C such that DL (g) ≥ C||[g − P(g)]||2σ ,

where ||.||σ denotes the following anisotropic norm  ||g||2σ ≡ ||(1 + |v|2 )γ/4 Πv ∇v g||2L2 +  

+||(1 + |v|2 )(γ+2)/4 [I − Πv ]∇v g||2L2 + ||(1 + |v|2 )(γ+2)/4 ||2L2

and where

Πv ∇g = (

v v .∇v g) . |v| |v|

From these results of Mouhot and coll., it follows that if γ + ν > 0 and ν > 0, then the resolvent of LB is compact, and thus the spectrum is purely discrete and the eigenvectors basis is complete. As shown by the results of Guo [76] and Mouhot and Neumann [110], this last result is one of the key ingredients to get global solutions for the spatially inhomogeneous Landau equations in the context of smooth perturbative solutions near equilibrium, see further details later on. On the other hand, the above result for the Bolztmann case is still not enough to perform a similar study. As we have already said, the computations of the formal symbol in the linear case shows that we are still missing mixed anisotropic and quite intricate norms, as already is the case for the Landau equation.

BOLTZMANN EQUATION

51

11. Spatially inhomogeneous Landau equation: Existence and regularity. We now turn to the spatially inhomogeneous Landau equation, so that we are looking for functions really depending on position variable x. Weak “renormalised” solutions were already constructed by Villani [128], but as usual, within this framework, it is still difficult to study fine properties of solutions. Small pertubative solutions were then constructed by Guo [76] for the Landau case, and also for Boltzman equation in the cutoff case in a series of works, [77, 78, 79, 80] on one hand, and by [97, 98] on the other hand. Since then, a number of papers have used these ideas, see for example [86, 60, 110]. 11.1. The perturbative framework of Mouhot and Neumann. We shall first present these results through the framework of Mouhot and coll. [110], which is general enough to cover a large class of collisional linear or nonlinear equations. However, we refer to the last subsection for some comments on this method, which certainly needs to be improved since at the moment, results of Guo and all. are certainly better. The general framework of [110] is as follows. Consider the following (nonlinear) equation ∂t f + v.∇x f = Q(f ), t ≥ 0, x ∈ Ω ≡ Tn

(163)

in dimension n and where we are working on the n dimensional torus. We assume there is a unique equilibrium on the torus, denoted by M , independent 1 from variables t and x. Considering the linearization f = M + M 2 h, one obtains the following linearized equation on h ∂t h + v.∇x h = L(h),

(164) 2

n

n

L being the corresponding linearized operator, and h being in L (T × R ). Standard L2 spaces will be used, together with the associated Sobolev space H k for the full (x, v) variables, indicating eventually the subscript x or v. Then the following assumptions are made in [110]. H1 ) Setting T ≡ L − v.∇x ,

(165)

it is assumed that L is a closed and self-adjoint operator, local w.r.t. variables t and x, such that L=K −Λ

(166) n

where Λ is a coercive operator: there exists a norm ||.||Λv on R (velocity space) such that

and

ν0Λ ||h||2L2v ≤ ν1Λ ||h||2Λv ≤ (Λ(h), h)L2v ≤ ν2Λ ||h||2Λv (∇v Λ(h), ∇v h)L2v ≥ ν3Λ ||∇v h||2Λv − ν4Λ ||h||2L2v

for some non negative constants νiΛ , and that

(L(h), h)L2v ≤ C L ||h|||Λv ||g||Λv .

52


Introduce also the norm also w.r.t. variable x ||.||Λ = ||||.||Λv ||L2x H2) Mixing property in velocity: assume that K has a regularizing effect in the following sense: for any δ > 0, there exists an explicit constant C(δ) such that for any h ∈ Hv1 , (∇v K(h), ∇v h)L2v ≤ C(δ)||h||2L2v + δ||∇v h||2L2v .

(167)

H3) Relaxation to the local equilibrium: we assume that L, as an operator in L2v , has a finite dimensional kernel N (L) = Span {φ1 , ..., φN } and we denote by Πl the orthogonal projection on N (L). It is assumed the following local coercivity assumption: there exists λ > 0 such that (L(h), h)L2v ≤ −λ||h − Πl (h)||2Λv . These assumptions imply that L is non positive and has a spectral gap in L2 by a size given by a lower bound ΛL . When variable x is added, Πl will denote the projection on the fluid part, while (Id − Πl ) is the projection on the kinetic part: Πl (h) =

N Z X ( i=1

hφi dv)φi

Rn

Since any local equilibrium, uniform in space, is in fact a global equilibrium, we note that Z d ∀i = 1, ..., N, hφi dxdv = 0 dt Tn ×Rn and so we denote Πg (h) =

N Z X ( i=1

hφi dxdv)φi

Tn ×Rn

which is time and space independent, and which is also the orthogonal projection on N (T ) in L2x,v . Then the first main result of [110] is concerned with the linearized equation Theorem 11.1. If T is a linear operator on L2x,v satisfying assumptions H1 to H3, 1 then it generates a strongly continuous evolution semi-group etT on Hx,v , which satisfies 1 ||etT (Id − Πg )||Hx,v ≤ CT exp[−τT t] for some explicit constants CT , τT . More precisely, for all h ∈ H k ,

(T h, h)H1 ≤ −CT′ (||h − Πg (h)||2Λ + ||∇x,v (h − Πg (h))||2Λ for some explicit Hilbert norm equivalent to the H 1 norm. The second main result of [110] is concerned with the non linear cutoff spatially inhomogeneous Boltzmann equation and Landau equation

BOLTZMANN EQUATION

53

Theorem 11.2. Consider either the Boltzmann equation with hard spheres or hard potentials with cutoff, or the Landau equation, with γ ≥ −2, on the torus. Let f0 ≥ 0 an initial datum in L12 . Denote by f∞ the unique equilibrium associated with f0 . Let k an integer such that [k/2] > N/2, and let the initial datum satisfy ||f −1/2 f0 ||H k ≤ ε

for some 0 < ε ≤ ε0 for an explicit constant ε0 . Then, there exists an unique global solution f = f (t, x, v) ≥ 0 in C([0, +∞[; H k ) of the Boltzmann or Landau equation, such that −1/2 [f (t, .., ) − f∞ ]||H k ≤ C exp[−τ t] ∀t ≥ 0, ||f∞

for explicit constants.

We shall not give the proofs, which is again unfortunate, and which are really interesting, and based on now well known ideas from Hypocoercivity [135, 136, 137, 138, 82, 83, 84, 85]. Let us just say that one estimates the time derivative of (separately) the gradients w.r.t. variables x and v, and use the assumptions. Then, and this is also one of the main important point, one also deals with the derivative of the mixed derivative (∇x h, ∇v h). In the end, one is able to construct a suitable functional , equivalent to the H 1 norm, and decreasing. Similar ideas are also used in the nonlinear setting, but one needs to deal with higher order norms. Let us just mention that in the Landau case, the explicit norm is given by ||h||2Λv = ||h < v >1+γ/2 ||2L2v + ||(P (v)∇v h) < v >1+γ/2 ||2L2v +||(I − P (v))∇v h) < v >γ/2 ||2L2v

as we have already seen in Section 10. This norm is stronger than L2v norm as soon as γ ≥ −2. As far as we known, similar results are still not available for the Boltzmann case without cutoff, but for the Maxwellian case, a recent work now is able to deal with this case, see [15]. In fact, the above norm in the Landau case is anisotropic, and that point explains why the situation is really difficult in the Boltzmann case. At least, one should be s able to show a lower bound in Hγ/2 , but up to now such a result has not be proven, except in the case γ = 0, that is the Maxwellian case! This particular point explains why [15] has succeeded in producing a positive result. 11.2. Regularization properties for Landau equation. Before turning to the regularization properties shown by Desvillettes and coll. [39], it is fair to recall the first result about global existence of perturbative solutions close to maxwellians due to Yan Guo Theorem 11.3. [76] Let γ ∈ [−3, 1] and N ≥ 8. Assume that F0 = F (0, ., .) satisfies Z Z √ F0 (x, v) µ(1, v, |v|2 )T dxdv = 0 (168) T3 ×R3

√ and that f0 = µ + µF0 is non negative. Then, there exist a constant κ0 > 0, such that if ||F0 ||Hx,v N ≤ κ0 , the spatially inhomogeneous Landau equation has a

54


unique global (in time) non negative classical solution f ≡ f (t, x, v). Moereover, there exists a constant C0 such that 1) If γ ∈ [−2, 1], then N ≤ C0 ||F0 ||Hx,v N . ||F ||L∞ t (0,+∞;Hx,v )

(169)

2) If γ ∈ [−3, −2], then

P supt∈(0,+∞) |α|+|β|≤N ||(∂xα ∂vβ F )(1 + |v|)(γ+2)|β| ||L2x,v P ≤ C0 |α|+|β|≤N ||(∂xα ∂vβ F0 )(1 + |v|)(γ+2)|β| ||L2x,v .

(170)

The regularization property of such solutions was then proven in [39] √ Theorem 11.4. Let γ ∈ [−3, 1] and f0 = µ + µF0 be non negative. Then there 8 exists a constant ε0 ∈]0, κ0 [, such that if ||F0 ||Hx,v ≤ ε0 , the unique nonnegative solution to the spatially inhomogeneous Landau equation given by Theorem 11.3 satisfies ∞ f ∈ Ct∞ ([τ, T ]; Hs,x,v (T3 × R3 )

(171)

for any 0 < τ < T < ∞ and any s ≥ 0 (this is the weighted space with parameter s). It is important to notice that the uniqueness and propagation of regularity of Guo’s solutions are used in their proof. It avoids the use of regularization operators, which are used in the context of Boltzmann equation, see further details later on. The proof of [39] uses in an essential way hypoelliptic results based on Bouchut’s method [27]. 11.3. Comments on perturbative solutions. Recent results about existence of such perturbative solutions for both Landau and Boltzmann equations with cutoff are in fact due to Guo [76, 77, 78, 79, 80] around a global Maxwellian, and Tai-Ping Liu and coll. [97, 98, 101, 102, 141, 142]. We have presented above the abstract framework of Mouhot and coll. which seems to encompass these results. In fact, this is not the case. First of all, note that the assumption that the position space is really the torus is important therein. Compare for instance to the results from [86] extending in particular the results of Guo [76] to the full position space Rn (instead of the torus Tn ). On the other hand, at least for the Boltzmann equation with cutoff assumption, no assumption of regularity w.r.t. velocity variables is required in the method of Guo, while it seems to be important in the abstract framework. Furthermore, macroscopic quantities are important in Guo’s approach. Therefore, it remains to clarify these points, which are also at the basis of the hypocoercivity method. Though this review is mainly concerned with singular kernels, we shall present the results taken from [79] related to Boltzmann equation with hard sphere cross section with cutoff, so that one may compare with the results of Mouhot and Neumann. Works related to Landau equation with the same method as Guo’s can be found in [86]. Finally, let us mention the work of Duan [60], avoiding the use of time derivatives in Guo’s method. We follow Guo’s notations very closely.

BOLTZMANN EQUATION

55

In particular, we use now the ω−representation, and consider Boltzmann’s equation in dimension 3: ∂t F + v.∇x F = Q(F, F ) where the cross-section is given by the hard sphere collision Z Z Q(F, F ) = dudω|(u − v).ω|{F (v ′ )F (u′ ) − F (v)F (u)

(172) (173)

R3 S 2

with the ω−representation

v ′ = v − [(v − u).ω]ω, u′ = u + [(v − u).ω]ω.

(174) 2

Introducing again the normalized global Maxwellian µ(v) = e−|v| , and defining the perturbation f (t, x, v) to µ by √ F = µ + µf, (175) one obtains the following standard equation for f ∂t f + v.∇x f + Lf = Γ(f, f ), where the linearized collision opertor L is given by 1 √ √ Lf = − √ {Q( µf, µ) + Q(µ, µf )} = ν(v)f − Kf, µ the collision frequency ν(v) being defined as Z 2 ν(v) = 2π du|v − u|e−|u|

(176)

(177)

(178)

R3

and K is a compact operator in L2 (R3 ). The nonlinear collision operator Γ(g, h) is finally defined as 1 √ √ Γ(g, h) = √ Q( µg, µh). (179) µ We known that L is nonnegative, ad furthermore, for fixed (t, x), the nullspace of L is given by √ √ √ N = span { µ, vi µ, |v|2 µ, 1 ≤ i ≤ 3}. (180) For fixed (t, x) and any function f = f (t, x, v), P is defined as the v−projection in L2v to the null space N . Thus f (t, x, v) can be decomposed as f (t, x, v) = P(f ) + {I − P}(f )

(181)

and Pf is called the hydrodynamic part, while the remainder is the microscopic part. The first step is to get macroscopic equations. By plugging the decompositon into (176), expanding Pf as a linear combination in the basis in (180) {a(t, x) +

3 X j=1

√ bj (t, x)vj + c(t, x)|v|2 } µ,

(182)

one can then derive the macroscopic equations for these coefficients. Introduce another basis (1 ≤ i < j ≤ 3) √ √ √ √ √ vi |v|2 µ, vi2 µ, vi vj µ, vi µ, µ (183) to get the following equations

∇x c = lc + hc ∂ 0 c + ∂bi = li + hi

(184) (185)

56


∂bj + ∂ j bi = lij + hij , i 6= j 0

∂ bi + ∂a = lbi + hbi

(186) (187)

0

∂ a = l a + ha (188) where lc (t, x), li (t, x), lij (t, x), lbi (t, x) and la (t, x) are coefficients w.r.t. (183) for the linear term −{∂t + v.∇x + L}{I − P}f while hc (t, x), hi (t, x), hij (t, x), hbi (t, x) and ha (t, x) are the corresponding coefficients for Γ(f, f ). Further notations are needed: < ., . > denotes the standard L2 inner product in 3 R × R3 for functions g(x, v) and h(x, v) in L2 , and we introduce also the weighted scalar product < g, h >ν ≡< ν(v)g, h > with ||.||ν for the corresponding norm. ||.|| is used for both L2 norms w.r.t. variable x or variables (x, v). If γ = [γ0 , γ1 , γ2 , γ3 ] is a multi-index, set ∂ γ = ∂tγ0 ∂xγ11 ∂xγ22 ∂xγ33 , (note that only derivatives w.r.t. variables (t, x) are considered), and introduce the instant energy as (where N is fixed and N ≥ 4) 1 1 X γ |||f |||2 (t) = ||∂ f (t)||2 2 2 γ≤N

and the dissipation rate as

|||f |||2ν (t) = ||{I − P}f (t)||2ν + 2

X

0 0, δ0 > 0 and M > 0 such that if 12 |||f |||2 (0) ≤ M , there exists a unique global √ solution f (t, x, v) to (176) such that F (t, x, v) = µ + µf (t, x, ) ≥ 0 and Z 1 δ0 t 2 |||f ||| (t) + |||f |||2ν (s)ds ≤ C0 |||f |||2 (0). 2 2 0

The paper [79] gives a very short and clear proof. The main points are as follows. Firstly, L is positive but only w.r.t. the microscopic part: there exists δ > 0 such that < Lf, f >≥ δ|L{I − P}f ||2ν (189) so that roughly speaking one controls only the microscopic part. Thus the main step is to control the macroscopic part, that is a, b and c. It is then noticed that the macroscopic equation (184) to (188) behave like an elliptic system, so that one can estimate the L2 norms of the derivatives of a, b and c. In a periodic setting, this is enough by using Poincare inequality. in the whole space, a little care is needed. Finally, the paper by Duan [60] shows how to avoid the use of time derivatives in the method of Guo. More informations on the macroscopic part can be found therein. Certainly, it could be applied to the Landau case considered in [86].

BOLTZMANN EQUATION

57

12. Spatially inhomogeneous Boltzmann equation: Introduction. We shall now, at last, begin to consider the spatially inhomogeneous Boltzmann equation, looking for solutions depending on both space and velocity variables. Up to the 2008’s, despite the recent progress made in the cutoff case mentioned previously, we are aware of five results in the non cutoff case: 1) the renormalised solutions (with defect measures), as introduced in [16]. This is parallel to the theory of renormalised solutions for Landau equation [128]. This aspect is well explained in [134], so we shall not comment further on this point. Let us just add that, even if we have a stronger notion of solutions, the renormalization approach completely justifies the Landau asymptotics [17]. As for the renormalised solutions in the cutoff case, many open problems remain open. 2) the result of Ukai [122] leading to a local existence result in Gevrey type space. 3) The special 1D space Boltzmann equation, for which a Bony functional is available. Then, see Cercignani, Villani [35, 36, 129], weak solutions are available. Let us mention also the recent work by Biryuk and all [25], in the cutoff case, leading to the existence of mild solutions, but it is not clear if this approach extends to the non cutoff case, and in fact one may be doubtful about such a possibility, in particular because of the use of mild formulations. 4) the result of Alexandre [7] showing in some cases the existence of solutions enclosed between travelling Maxwellians, see below for further details. 5) and finally the “Hong-Kong” team results, which after some preliminary works related to hypoellipticity effects, yields an existence and a regularizing result. Last point 5) will be discussed more precisely later on, and we shall begin our discusssion with Point 4). 13. Inhomogeneous Boltzmann equation: “Travelling” solutions. Some of the results of this Section are based on initial works in the cutoff case [81, 73], our work [7] and some results of the previous Sections. Additional results are available in [7]; in particular, the case of a perturbation by −ε∆v is considered therein, as well as other cases of collisional kernels. Furthermore, the results therein were stated in 3 dimensional case, but here, we shall consider any dimension, and for simplicity we shall only consider the case of small singularities 0 < ν < 1. We look for a function f = f (t, x, v), t ∈ R+ (or in (0, T ), with T > 0 fixed), (x, v) ∈ R2n , solution in a suitable sense of problem (1). We are now working with the σ-representation. We assume that f0 = f0 (x, v) is a given initial datum and that it satisfies −

m− M (0, x, v) ≤ f0 (x, v) ≤ m+ M (0, x, v),

(190)

+

where 0 < m ≤ m are given constants and

2

2

M (t, x, v) = e−|v| e−|x−tv| .

−

(191)

The assumption m > 0 is not necessary for the existence results presented below, but it simplifies the regularity questions dealt with. Our aim is to show that there exists T > 0 so that Boltzmann equation admits weak solutions in the class L1 ∩ L∞ ((0, T ) × R2n x,v ), for initial data satisfying (190), and this will hold true for any value of m+ . The above comparison assumption is classical, and we recall that the function M is a special solution of problem (1) as Q(M ) = 0 and ∂t M + v.∇x M = 0.

58


Such studies already exist in the cutoff (or non singular) case, but here we deal with the non cut-off (singular) case. As usual, we shall also make the assumption that our kernel B is given by a multiplicative form, γ denoting the exponent of the kinetic part, and ν the exponent of the angular part B(|v − v∗ |, cos θ) = |v − v∗ |γ b(cos θ), sinn−2 θb(cos θ) ∼θ→0 Kθ−1−ν .

(192)

Under the above assumptions, we shall say that, for T > 0, possibly T = +∞, f ≥ 0 is a weak solution of problem (1) if f ∈ L∞ (0, T ; L1 ∩ L∞ (R2n )), satisfies the usual entropic bounds, together with a bounded entropy dissipation rate, and the Boltzmann equation in weak sense. The fact that < Q(f ); h > for (at least) such h as defined above is meaningfull should now be clear, in view of previous Sections. 13.1. An existence result. The main result adapted from [7] is given by Theorem 13.1. Let the collision kernel be given by (192). Assume that γ < −1. Then, for all m+ > 0, there exists T ∈ R+∗ and two non negative C 1 functions δ, β on [0, T ], with δ(0) = m− and β(0) = m+ , such that Boltzmann equation admits a weak solution f satisfying δ(t)M (t, x, v) ≤ f (t, x, v) ≤ β(t)M (t, x, v).

Moreover, if γ ≥ 1 − n, then T = +∞ is allowed. The solutions constructed above satisfy ν hf ∈ L2 (0, T ; H 2(1+ν) (R2n ) for all h ∈ C0∞ (]0, T [×R2n ). For the regularity result, two facts are only needed: that a solution is bounded in L1 ∩ L∞ , but also that it is bounded from below by a strictly positive function which is also in L1 ∩ L∞ . Of course, this is the case in particular for the solutions constructed by the first result. As far as we know, this is still the first existence result for the spatially inhomogeneous Boltzmann equation without cutoff, without assuming any regularity assumptions on the initial data. However, it is limited to soft potentials. Let us give the main lines of the proof of Theorem 13.1. The first step consists in a study of a cutoff problem. At this point, previous works do not apply, since the constants therein depend on the cutoff assumptions. Fixing a parameter n′ ∈ N∗ , large enough, consider the following cutoff type problem ∂t f + v.∇x f = Qn′ (f, f ), (193) f |t=0 = f0 , where Qn′ is the non singular Boltzmann operator corresponding to the non singular cross section Bn′ = BIsin θ≥ 1′ . (194) n

We want to find T > 0, two C 1 functions δ, β : [0, T ] → R+∗ , δ(0) = m− , β(0) = m+ , which do not depend on n′ , and such that problem (193) admits a ′ weak solution f (= f n ) such that δM ≤ f ≤ βM . 0 For any 0 < T ≤ +∞, δ and β given functions in C+ ([0, T ]), and l a fixed and given function satisfying δ(t)M ≤ l ≤ β(t)M, (195)

BOLTZMANN EQUATION

59

we introduce the following linear problem Z Z Z Z  ′ ′  ′ ∂ g + v.∇ g = B (., .)l l − Bn′ (., .)l∗′ g  x n ∗   t n n−1 n n−1 RZ SZ R S ′ ′ (., .)(l − l∗ ), +g B  n ∗   Rn S n−1  g|t=0 = f0 .

Then one has, for all time t Z Z Rn

(196)

Bn′ (., .)l′ l∗′ ≤ Cβ 2 (t)

S n−1

for some constant depending on n′ . This shows that for all T > 0 fixed, problem (196) admits an unique solution g (in the mild and distributional sense) which is in L1 ∩ L∞ ((0, T ) × R2n x,v ) and ≥ 0. Next, we look for an upper solution of problem (196) in the form a ≡ β(t)M . More precisely, we wish to find a sufficient condition on β for this purpose. After some computations, we are led to look for β such that (β ≥ 0)  Z Z Z Z  ′ Bn′ (., .)(l∗′ − l∗ ), β (t)M ≥ β Bn′ (., .)l∗′ (M ′ − M ) + βM (197) n n−1 n n−1 R S β(0) ≥ m+ . R S Note that the right hand side is nothing else than βQn′ (l, M ). Firstly, by the Cancellation Lemma 4.1, one has Z Z Z Bn′ (., .)(l∗′ − l∗ ) = Sn′ (| v − v∗ |)l∗ dv∗ , Rn S n−1

(198)

Rn

with the bound | Sn′ (| v − v∗ |) |≤ c1 | v − v∗ |γ

(199)

′

where c1 does not depend on n . Then, one can show that Z

1 . (1 + t)n+γ In conclusion, it is enough to choose β ≥ 0 such that  Z Z 1  ′ β (t)M ≥ Bn′ (., .)l∗′ (a′ − a)dσdv∗ + β 2 (t)M c2 , (1 + t)n+γ n n−1 β(0) ≥ m+ ,R S R R For the most difficult term Rn S n−1Bn′ (., .)l∗′ (a′ − a)dσdv∗ , one finds Sn′ (| v − v∗ |)l∗ dv∗ ≤ β(t)c2

Rn

|

Z Z Rn

Bn′ (., .)l∗′ (a′ − a)|dσdv∗ ≤ β 2 (t)M

S n−1

(200)

(201)

1 . (1 + t)n+γ

By the above computations, it is enough to choose β ≥ 0 such that ′ β (t) ≥ β 2 (t)c3 (1+t)1 n+γ , β(0) ≥ m+ .

(202)

Once we get at this point, looking for a lower solution in the form δ(t)M on the same time interval [0, T ] is classical.

60


We have therefore achieved the following: for γ < −1, there exists T > 0 and β, δ C 1 functions from [0, T ] in R+∗ , β(0) = m+ , δ(0) = m− , which do not depend on n′ , such that for all l ∈ L1 ∩ L∞ ((0, T ) × R2n x,v ), δM ≤ l ≤ βM, problem (196) has an unique solution g such that δM ≤ g ≤ βM.

If moreover, γ ≥ 1 − n, then we can take T = +∞. By a classical fixed point argument, we can assert that there exists gn solution of the following Boltzmann equation with cutoff ∂t gn + v.∇x gn = Qn (gn ), (203) gn |t=0 = f0 , on the time interval [0, T ], such that δM ≤ g n ≤ βM , where T , δ and β are as above, not depending on n. Furthermore, gn satisfies the following uniform entropic dissipation rate bound estimate Z TZ Z Z g′ g′ ′ − gn gn∗ } ln n n∗ dσdv∗ dvdxdt ≤ CT , Bn′ (., .){gn′ gn∗ (204) gn gn∗ 0 R2n Rn S n−1 as it is clear by multypling (203) by ln g n . The second step of the proof consists then in sending n′ to +∞. From (204) and the (uniform in n′ ) L∞ bound on gn′ , one deduces that Z TZ 0

Z

R2n

Rn

Z

S n−1

Bn′ (., .){gn′ ′ gn′ ′ ∗ − gn′ gn′ ∗ }2 dσdv∗ dvdxdt ≤ CT .

This is enough to apply known results and arguments from renormalised framework, see for example [16]. In particular, there exists f ∈ L1 ∩ L∞ , δM ≤ f ≤ βM such that (for a suitable sub-sequence) gn′ → f in Lp ((0, T ) × R2n x,v ) strongly (1 ≤ p < +∞). Writing the distributional formulation associated to (203), it follows that f is a weak solution, ending the proof. Finally, the regularization property of such solutions uses the coercivity results from Section 5, together with a nice argument introduced by Perthame [114] for getting a small regularization effect, on which we will have more to say, using different tools. 13.2. Maximum type principles. The previous existence result is in fact based upon hidden maximum type principles, see for example [69] in the non cutoff case, and for earlier references. For a given function l = (t, x, v) smooth enough, consider the linear Boltzmann equation ∂t f + v.∇x f = Q(l, f ).

(205)

Assume that l is positive. How is it then possible to show that f is positive? This type of questions is important for instance if one uses an iteration scheme to show existence result. We depart from [69], and instead present another proof based on renormalisation principles [16].

BOLTZMANN EQUATION

61

Let α(t) be a smooth convex function. Then forgetting integration w.r.t. variable x, one has Z Z ZZ dvQ(l, f )α′ (f ) = dv dv∗ dσB(.)l∗′ (f ′ − f )α′ (f ) Z ZZ ′ + dvf α (f ) dv∗ dσB(.)(l∗′ − l∗ ).

Using the Cancellation lemma 4.1, it follows that Z Z ZZ Z ′ ′ ′ ′ dvQ(l, f )α (f ) = dv dv∗ dσB(.)l∗ (f − f )α (f ) + dvf α′ (f )S ∗v l(v). Since we know that l ≥ and that α is convex, it follows that Z

′

dvQ(l, f )α (f ) ≤

Z

dv

ZZ

dv∗ dσB(.)l∗′ (α(f ′ )

ZZ

dv∗ dσB(.)l∗′ α(f ′ )

− α(f )) +

Z

dvf α′ (f )S ∗v l(v).

Z

dvf α′ (f )S ∗v l(v)

Therefore, by adding and substracting an additional term, one gets Z

′

dvQ(l, f )α (f ) ≤

Z

dv

− Z

Z

α(f )

ZZ

− l∗ α(f ) +

B(.)[l∗′ − l∗ ].

Thus again, using the Cancellation Lemma, it follows that Z ZZ Z ′ ′ ′ dvQ(l, f )α (f ) ≤ dv dv∗ dσB(.)l∗ α(f )−l∗ α(f )+ dv[f α′ (f )−α(f )]S ∗v l(v). We note that the first term of the r.h.s. of this inequality is zero, and therefore Z Z dvQ(l, f )α′ (f ) ≤ dv[f α′ (f ) − α(f )]S ∗v l(v).

Getting back to the (linear) equation (205) satisfied by f , one gets ZZ ZZ d dxdvα(f ) ≤ dxdv[f α′ (f ) − α(f )]S ∗v l(v). dt Now, let α(t) = 21 (t− )2 . Then, one gets exactly ZZ ZZ d dxdvα(f ) ≤ dxdv[α(f )]S ∗v l(v). dt Thus, if we know that say S ∗v l is uniformly bounded, this is the case for example if γ ≤ 0 and l is bounded by a travelling Maxwellian, then one gets the positivity of f , if the initial data f0 is positive. Can we use this type of ideas for extending our previous result on the existence of solutions enclosed between travelling Maxwellians? Thus, instead of considering the linear equation of the previous sub-section we use this time (205), but still assuming δ(t)M ≤ l ≤ β(t)M . Then h ≡ β(t)M − f satisfies ∂t h + v.∇x h = β ′ (t)M − Q(l, f ) that is ∂t h + v.∇x h = β ′ (t)M + Q(l, h) − β(t)Q(l, M ) The previous arguments with α(t) = 21 (t− )2 , shows that if we want h ≥ 0, it is enough to ask for

62


ZZ

dxdv[β ′ (t)M − β(t)Q(l, M )]α′ (h) ≤ 0

and therefore enough to ask that punctually [β ′ (t)M − β(t)Q(l, M )] ≥ 0. Thus, we are back to the computations of the previous subsection. We did not improve anything but only simplify the first steps! 14. Hypoelliptic properties and kinetic equations. We have seen that in the spatially homogeneous case, elliptic properties for solutions to Boltzmann equation may be expected. We have seen in the previous Section 13 that a small hypoelliptic effect is true for some solutions to the spatially inhomogeneous Boltzmann equation. We have also seen that full hypoelliptic effects were also true for the spatially inhomogeneous Landau equation, see Section 11. Let us recall also that at the level of renormalised solutions, strong compactness is available, see [9, 16, 94, 95, 128]. We therefore expect that something similar will hold true for a larger class of cross-sections for Boltzmann equation. As a first step, we need to have an hypoelliptic tool which will be easy to use. Recall that in Section 11, the strategy of Bouchut [27] was used for Landau equation. This Section is based on our recent paper [13]. We shall make contact with the uncertainty principle which was introduced by Heisenberg, [63] for example. We shall see how to apply this principle in the framework of spatially inhomogeneous kinetic equations. As far as we know, though this principle was applied for many partial differential equations, including Schrödinger type equations, cf. [62] and many subsequent works (see a more extended bibliography in [13]), it seems this is the first time that ideas related to the uncertainty principle are applied in the context of kinetic equations. 14.1. A model problem. Before turning to the results from [13], let us first of all describe the results of Morimoto and Xu [106], where they consider hypoelliptic effects associated with the equation ˜ v )α f = g, P f = ∇t f + v.∇x + σ0 (∆

(206) ˜ v ) ≡ |D ˜ v |2α where (x, v) ∈ R , and σ0 > 0 is a given constant. Furthermore (−∆ 2α 2 ˜ if |ξ| ≥ 2 and to |ξ| if |ξ| ≤ 1. is a Fourier multiplier with a smooth symbol |ξ| If α = 1, this is a linear Fokker Planck equation studied recently by Bouchut [27], where it proven the maximal hypoellipticity of order 2/3, but see also numerous works related to previous works of Hormander, see [136, 137, 138] and the references therein for example. Assuming now that α < 1, one then notices that the above operator is not a standard one. This is the motivation for the study of Morimoto and Xu [106], to give a very short proof of hypoelliptic effects, having in mind applications to the Boltzmann case. Indeed, the above model can be thought as a very simplified Boltzmann model. If one assumes that g ∈ L1 (0, T ; H s (R2n )), for some 0 < T < +∞, s ≥ 0, and that f0 ∈ H s (R2n ), then one can show that the asssociated Cauchy problem with initial data f0 has a unique weak solution f , such that 2n

˜ v )α/2 f ∈ L2 (0, T ; H s (R2n )) f ∈ L∞ (0, T ; H s(R2n )), (−∆

The main hypoelliptic result from [106] is given by

α

BOLTZMANN EQUATION

63

Theorem 14.1. Assume that 1/3 < α < 1, s ≥ 0 and g ∈ H s (]a, b[×R2n )) and f ∈ L2 (]a, b[×R2n ) is a weak solution of P f = g on ]a, b[×R2n . Then there exists k0 ∈ N such that α

1

< v >−k0 −1 f ∈ H s+ 4 − 12 (]a′ , b′ [×R2n ) for all a < a′ < b′ < b. In particuliar if g ∈ H ∞ , then u ∈ H ∞ .

Above k0 is of the order [4s(α − 31 )−1 ] + 1. So the values of α less that 1/3 cannot be tackled by the method of Morimoto and Xu, and also arises the problem of good estimates w.r.t. weights. Without giving too much details, we want just to describe the main ideas of the ˜ v )α on an open domain Ω of the full time, proof. Setting P = ∂t + v.∇x + σ0 (−∆ space and velocity variables, note that P is not a pseudo-differential operator. Let 1 Λ = (1 + |Dt |2 + |Dv |2 + |Dx |2 ) 2 and X0 = Λ−1/3 (∂t + v.∇x ), , Xj = Λ−α ∂vj , j = 1, ..., n.

α Then Xj ∈ Op(S1,0 (R2n+1 ), for j = 1, ..., n, but not X0 , since the coefficient v is unbounded. One important estimation is then given by the following result, reminiscent of Kohn’s technics

Lemma 14.2. Assume Ω is an open domain of R2n+1 and that 1/3 < α < 1. Then there exists a constant C > 0 such that n X ||Xj v||2L2 ≤ C{Re (P v, v) + ||v||2L2 } (207) j=1

for all v ∈

C0∞ (Ω),

and

||X0 v||2L2 ≤ C{ where

3 X

k=1

Re (P v, Ak v) + ||v||2L2 }

(208)

0 (R2n+1 ), A2 = (Λ−1/3 + Λ−1 )X0 , A3 = −(∂t + v.∇x )Λ−1 X0 . A1 ∈ Op(S1,0

This Lemma, together with Hormander-Kohn condition on the family X0 , ..., Xn gives the following sub-elliptic estimates Lemma 14.3. With the same notations as in Lemma 14.2, one has n X j=1 α

α

1

||Λ 2 − 6 −1 ∂x f ||2L2 ≤ C{ 1

||Λ 4 − 12 −1 ∂t f ||2L2 ≤ C{ and α

1

||Λ 4 − 12 f ||2L2 ≤ C{

3 X

k=1

3 X

k=1

3 X

k=1

Re (P f, Ak f ) + ||f ||2L2 }

Re (P f, Ak f ) + || < v > f ||2L2 }

Re (P f, Ak f ) + || < v > f ||2L2 }.

(209)

(210)

(211)

Similar estimates with weights are also needed and we refer to the proofs in [106]. That also explains why this is a local hypoellipticity result. Finally, let us note that Gevrey regularity is studied for some linear kinetic equations therein.

64


14.2. An uncertainty principle. In order to extend the previous results to more general kinetic equations, we now introduce a generalized version of the uncertainty principle. Let a+ and a− be two non-negative functions of variable v ∈ Rn , with a− in ∞ L (Rn ). Set as (Dv ) = |Dv |2s for 0 < s < 1, and as,log (Dv ) = (log < Dv >)2s for s > 1/2. In the spirit of Fefferman’s uncertainty principle, cf. [62], we wish to prove the following type of coercivity estimate under some reasonable conditions on the functions a+ and a− : there exists a constant c > 0 such that Re((a(Dv ) + a+ )f, f )L2 (Rn ) ≥ c(a− f, f )L2 (Rn )

for all f ∈ S(Rn ),

(212)

where a(Dv ) can be either as (Dv ) or as,log (Dv ). For this purpose, we need to introduce some notations. For the variable r > 0, we set ˜ r|2s−1 if a = as,log , a ˜(r) = r−2s if a = as , and a ˜(r) = |log

˜ r = log r for 0 < r ≤ 1, and log ˜ r = 0 otherwise. where log n For a given cube Q in R , we denote its side length by l(Q). Let Q∗ be any cube such that Q ⊂ Q∗ with l(Q∗ ) = 2l(Q). For each pair of such cubes Q and Q∗ in Rn , we define the following subset E(Q, Q∗ ) = v ∈ Q∗ ; a+ (v) ≥ ||a− ||L∞ (Q) − a ˜(l(Q)) . (213)

The main assumption on the functions a+ and a− is that there exists a uniform constant κ > 0 such that |E(Q, Q∗ )| ∗ ∗ (H0 ) inf ; all cubes Q ⊂ Q with 2l(Q) = l(Q ) ≥ κ, |Q∗ | where |E| stands for the Lebesgue measure of a set E ⊂ Rn , and the infimum is taken over all pairs of cubes Q and Q∗ in Rn satisfying the above property. Then, one gets [13]

Theorem 14.4. Uncertainty Principle Let a− ∈ L∞ (Rn ) and a+ , a− ≥ 0. If Condition (H0 ) holds for some κ > 0, then there exists a constant C > 0 depending only on κ and n such that Z Z 1 a− (v)|f (v)|2 dv ≤ C |as2 (Dv )f (v)|2 + a+ (v)|f (v)|2 dv, (214) Rn

Rn

1

for any f ∈ S(Rn ). The same coercivity estimate holds also when as2 ≡ |Dv |s is 1

2 replaced by as,log ≡ (log < Dv >)s with s > 12 .

Let us give the main ideas of the proof. For k ∈ Z, we consider the lattice Λk = 2−k Zn formed by the points in Rn whose coordinates are multiples of 2−k . Let Dk be the collection of cubes with side length ∞ 2−k and vertices located at the points in Λk . The cubes belonging to D = ∪ Dk −∞

are called dyadic cubes. If a cube Q ∈ D is obtained by dividing of a bigger dyadic cube Q′ with double side length, we call Q′ the mother cube of Q. For a constant A > 0 and a function 0 ≤ a− (v) ∈ L∞ (Rn ), a cube Q will be said to satisfy property F (A, a− ) if ||a− ||L∞ (Q) ≤ A˜ a(l(Q)).

BOLTZMANN EQUATION

65

Furthermore, we denote by M(A, a− ) the set of dyadic cubes Q ∈ D satisfying F (A, a− ) but not their mother cubes. Then, note that if a cube Q′ satisfies F (A, a− ), then so does any cube Q ⊂ Q′ . Since a− ∈ L∞ , all dyadic cubes in Dk with k large enough satisfy F (A, a− ). Moreover, the collection of cubes in M(A, a− ) is a non-overlapping covering on Rn , that is Rn =

∪

Q∈M(A,a− )

Q,

|Qk ∩ Qj | = 0, Qk , Qj ∈ M(A, a− ), k 6= j.

(215)

For the non-overlapping covering M(A, a− ), we now construct another locally ˜ uniform finite covering M(A, a− ) on Rn by taking into account Hypothesis (H0 ). This is in fact the main argument of the proof of the above Theorem Lemma 14.5. Under the hypothesis of Theorem 14.4, there exists a constant As depending only on s, such that for all A ≥ As , and any cube Qj ∈ M(A, a− ), there ˜ j satisfying the following conditions: exists another cube Q ′ ˜ (1) Qj ⊂ Qj ⊂ Qj , where Q′j denotes the mother cube of Qj ; (2) There exists a dual pair of cubes (Q, Q∗ ), Q having the side length 21 l(Qj ) ˜ j ∩ E(Q, Q∗ ), one has such that, by setting Ej = Q |Ej | ≥

κ |Qj |, 2

(216)

and a+ (v) ≥ C1 a ˜(l(Qj )) ≥ C2 ka− kL∞ (Qj ) , for any

v ∈ Ej .

(217)

˜ j } = M(A, f (3) The collection {Q a− ) is a locally finite covering of Rn , and the −n number of overlapping is order O(κ ) which is uniform in Rn . Finally, to finish the proof, except for the case s = 1, note that Z ZZ 1 |f (v) − f (w)|2 a ˜(|v − w|) 2 2 |as (Dv )f (v)| dv ≥ c1 dvdw |v − w|n n Rn Rn v ×Rw v ZZ |f (v) − f (w)|2 a ˜(|v − w|) ≥ c2 dvdw n P ˜ P ˜ |v − w| j Qj × k Qk Xa ˜j ) Z Z ˜(l(Q ≥ c2 |f (v) − f (w)|2 dvdw, ˜j | ˜ ˜ | Q Qj ×Qj j

where c1 , c2 are positive constants depending only on κ, n and s. In the case s = 1, the above inequality also holds because for any cube Q, we have ZZ Z 1 RR 2 2 |v − w| |∇u(v + θ(w − v)|2 dθ Q×Q |f (v) − f (w)| dvdw ≤ Q×Q 0 Z Z 1/2 Z 2 2 ≤ l(Q) dθ |∇f (v + θ(w − v)| dv dw Z Q Z QZ 01 +l(Q)2 dθ |∇f (v + θ(w − v)|2 dw dv. Q

1/2

Q

By the change of variables z = v + θ(w − v)R ( v → z, w → z respectively), the last two terms can be estimated by 2n l(Q)2 |Q| Q |∇f (z)|2 dz.

66


It follows from Lemma 14.5 that ˜ j ) RR a ˜ (l(Q 2 ˜ j ×Q ˜ j |f (v) − f (w)| dvdw ˜j | Q |Q ZZ c0 a ˜(l(Qj )) |f (v) − f (w)|2 dvdw ≥ ˜j | ˜ |Q Z Z Qj ×Ej c0 a ˜(l(Qj )) 1 2 2 ≥ |f (v)| − |f (w)| dvdw ˜j | 2 ˜ j ×Ej |Q Q Z Z c0 a ˜(l(Qj ))|Ej | ≥ |f (v)|2 dv − c0 a ˜(l(Qj )) |f (w)|2 dw ˜j| ˜j 2|ZQ Q E j Z c0 κ c0 ≥ n+2 a− (v)|f (v)|2 dv − a+ (w)|f (w)|2 dw, 2 C Ej Qj which leads to the desired estimate by summing over the index j. This completes the proof of Theorem 14.4. 14.3. Some applications of the uncertainty principle. As a first example, consider the following transport equation ft + v · ∇x f = g ∈ D′ (R2n+1 ),

(218)

where (t, x, v) ∈ R2n+1 . Then, one has Theorem 14.6. Regularity for transport equations ′ Assume that g ∈ H −s (R2n+1 ), for some 0 ≤ s′ < 1. Let f ∈ L2 (R2n+1 ) be 1

a weak solution of the kinetic equation (218) such that as2 (Dv )f ∈ L2 (R2n+1 ) for some 0 < s ≤ 1. Then it follows that s(1−s′ )/(s+1)

Λx f , (1 + |v|2 )ss′ /2(s+1) ′

s(1−s′ )/(s+1)

Λt f ∈ L2 (R2n+1 ). (1 + |v|2 )s/2(s+1) 1

2 (Dv )f ∈ L2 (R2n+1 ) Similarly, if g ∈ H −s (R2n+1 ), for some 0 ≤ s′ < 1, and as,log for some s > 1/2, it follows that

(log Λx )s−1/2 f , (1 + |v|2 )s′ /2

(log Λt )s−1/2 f ∈ L2 (R2n+1 ), (1 + |v|2 )1/2

where Λ• = (1 + |D• |2 )1/2 . The main idea in the proof of this theorem is to define the corresponding functions a± (v), entering the uncertainty principle, and show that they satisfy Condition (H0 ) with the Fourier variables τ and η being parameters. The different weights in the variable v for the differentiations w.r.t. the time and space respectively arise naturally from the form of the transport operator. Similar results were obtained with different methods by Bouchut in [27], that is, by using Fourier transform, Hörmander’s type commutators and velocity averaging techniques, see more details and complete bibliography in [29]. In particular, we refer to Theorem 1.5 in [27]. However, notice that the results in [27] do not include the above general Theorem 14.6 because in particular the differentiation of integer order w.r.t. the variable v is required in [27]. Furthermore, we avoid using the velocity averaging argument but rely on a newly proved generalized uncertainty principle.

BOLTZMANN EQUATION

67

The main lines of the proof are as follows. Using the uncertainty principle, the Theorem will be proven once the following estimate is obtained n 1 kA(v, Dt , Dx )f k2L2 (R2n+1 ) ≤ C kas2 (Dv )f k2L2 (R2n+1 ) (219) o +kX0 f k2H −s′ (R2n+1 ) + kf k2L2 (R2n+1 ) ,

for all f ∈ S(R2n+1 ), where X0 ≡ (∂t + v · ∇x ) and ′

′

< Dx >s(1−s )/(s+1) f < Dt >s(1−s )/(s+1) f + . A(v, Dt , Dx )f = ′ (1 + |v|2 )ss /2(s+1) (1 + |v|2 )s/2(s+1)

Letting τ , η and ξ denote the dual (Fourier) variables corresponding to t, x and v respectively, we use standard notations from the pseudo-differential operator theory. 2 2 Choose χ(τ, η, ξ) ∈ S 0 , such that χ = 1 on Γ = {(τ, η, ξ) ∈ R1+2n τ,η,ξ ; |τ | + |η| ≥ ˜ = {|τ |2 + |η|2 ≤ 1 + |ξ|2 }. Denote by χ(D) the 1 + |ξ|2 /2 }, and χ = 0 outside of Γ pseudo-differential operator with symbol χ. We have kA(v, Dt , Dx )f kL2 (R2n+1 ) and one can check that

≤ +

kA(v, Dt , Dx )χ(D)f kL2 (R2n+1 )

kA(v, Dt , Dx )(1 − χ(D))f kL2 (R2n+1 ) , 1

kA(v, Dt , Dx )(1 − χ(D))f kL2 (R2n+1 ) ≤ ||as2 (Dt )(1 − χ(D))f ||L2 (R2n+1 ) 1

1

+||as2 (Dx )(1 − χ(D))f ||L2 (R2n+1 ) ≤ 2||as2 (Dv )(1 − χ(D))f ||L2 (R2n+1 ) 1

Since

≤ C(||as2 (Dv )f ||L2 (R2n+1 ) + ||f ||L2 (R2n+1 ) ) . ′

k[< D >−s X0 , χ(D)]f kL2 (R2n+1 ) ≤ C||f ||L2 (R2n+1 ) , it is therefore enough to prove that 1 kA(v, Dt , Dx )χ(D)f k2L2 (R2n+1 ) ≤ C kas2 (Dv )χ(D)f k2L2 (R2n+1 ) +kX0 χ(D)f k2H −s′ (R2n+1 ) + kf k2L2 (R2n+1 ) .

(220)

˜ we deduce that As supp Ft,x,v (χ(D)f ) ⊂ Γ, ! Z ZZ |τ + v · η|2 |Ft,x (χ(D)f )(v, τ, η)|2 dτ dη dv (1 + τ 2 + |η|2 )s′ Rn Rn+1 τ,η v ≤ ||X0 χ(D)f ||2H −s′ (R2n+1 ) ,

where F• denotes the Fourier transform with respect to the variable •. 1/2 Since the operator as (Dv ) is independent of t and x variables, (220) will follow if we can prove that c(a− (v, τ, η)u, u)L2 (Rn ) ≤ Re (a(Dv ) + a+ (v, τ, η))u, u L2 (Rn ) , u ∈ S(Rn ), (221) for any (τ, η) ∈ R1+n when |τ | , |η| ≥ R0 for some large constant R0 > 0. Here, a− (v, τ, η) = c0

′

′

|η|2s(1−s )/(s+1) |τ |2s(1−s )/(s+1) − + c = a− 0 1 (v, τ ) + a2 (v, η), (1 + |v|2 )s/(s+1) (1 + |v|2 )ss′ /(s+1)

for some constant c0 > 0, and

a+ (v, τ, η) = 1 +

|τ + v · η|2 . (1 + τ 2 + |η|2 )s′

68


We can check the Condition (H0 ) for the uncertainty principle given in (221) when (τ, η) ∈ R1+n is considered as some parameter such that |τ | , |η| ≥ R0 , for some large constant R0 > 0. Similarly, Condition (H0 ) is also satisfied for the functions a+ (v, τ, η), a− 1 (v, τ ) and a− (v, η). 2 Finally, we have obtained kas1/2 (Dv ) χ(D)f k2L2 (R2n+1 ) + kX0 χ(D)f k2H −s′ (R2n+1 ) Z Z Z ′ |τ |2s(1−s )/(s+1) ≥ c |Ft,x (χ(D)f )(v, τ, η)|2 dτ dη 2 )s/(s+1) (1 + |v| n n Rv R ZZ ′ |η|2s(1−s )/(s+1) 2 |F (χ(D)f )(v, τ, η)| dτ dη dv, + c t,x 2 ss′ /(s+1) Rn (1 + |v| ) and this completes the first part of Theorem 14.6. The following result follows, involving weighted spaces. Corollary 14.7. Let l ∈ N and δ > 0 such that 1 − s′ − δ > 0. Then, for any value M (δ, l) ≥ (l + δ)(1 − s′ )/δ, one has n 1 e Dt , Dx ) < v >l f k2 2 2n+1 ≤ C kas2 (Dv ) < v >l f k2 2 2n+1 (222) kA(v, L (R ) L (R ) o 2 M(δ,l) 2 +kX0 f kH −s′ (R2n+1 ) + k < v > f kL2 (R2n+1 ) ,

for any f ∈ S(R2n+1 ), where < v >= (1 + |v|2 )1/2 , X0 ≡ (∂t + v · ∇x ) and ′

′

s(1−s −δ)/(s+1) < Dt >s(1−s −δ)/(s+1) e Dt , Dx ) = < Dx > A(v, + . ′ /2(s+1) 2 ss (1 + |v| ) (1 + |v|2 )s/2(s+1)

As a second application of the Uncertainty principle, consider the linearized Boltzmann equation around a global Maxwellian. More precisely, we linearize the Boltzmann equation around the normalized Maxwellian distribution |v|2 3 µ(v) = (2π)− 2 e− 2 , where we have restricted ourselves to the physical case where the space (velocity) dimension equals to 3, although it is straigtforward to extend the argument to any dimension greater than 2. The collision cross section will be assumed to be of the form

v − v∗ π B(|v − v∗ |, cos θ) = Φ(|v − v∗ |)b(cos θ), cos θ = ,σ , 0≤θ≤ . |v − v∗ | 2 We will only consider Maxwellian molecule type case, that is, we take Φ = 1 and B = b(cos θ) ≈ Kθ−2−ν , when θ → 0+,

(223)

Lf = Q(µ, f ) + Q(f, µ).

(224)

and we will also restrict to the case of small singularities 0 < ν < 1. Since µ is the global equilibrium state satisfying Q(µ, µ) = 0, the bilinearity of Q(g, f ) yields Q(µ + g, µ + g) = Q(µ, g) + Q(g, µ) + Q(g, g). Thus, the linearized Boltzmann operator takes the form Consider the following Cauchy problem for the linearized Bolztmann equation ft + v · ∇x f = Lf, x, v ∈ R3 , t > 0 ;

f |t=0 = f0 .

(225)

BOLTZMANN EQUATION

69

It may be shown, see [13] for details, that if we assume that < v >l f0 ∈ L2 (R3 × R3 ), for any l ∈ N, then there exists a unique weak solution f of the linearized Boltzmann equation (225) such that < v >l f ∈ L∞ ∩ L2 (]0, T [×R6 ), for any l ∈ N and for any T > 0.

Here, we require that all the moments are bounded in L2 (R6 ) just to ensure that the solution becomes H +∞ w.r.t. all variables for any positive time as asserted by Theorem 14.8. Full regularity for Linearized Boltzmann equation Assume that f is a weak solution of the linearized Boltzmann equation such that < v >l f ∈ L2 (]0, T [×R6 ), for all l ∈ N.

Then, for any l ∈ N and for any 0 < δ < T ′ < T , we have < v >l f ∈ H +∞ (]δ, T ′ [×R6 ).

The proof of this result will not be given here, and we refer to [13] for details. Some of the tools which are used will be mentioned later for the spatially inhomogeneous nonlinear Boltzmann equation. As a third application, we consider the spatially inhomogeneous Boltzmann equation, ft + v · ∇x f = Q(f, f ), x, v ∈ R3 , t > 0 ; f |t=0 = f0 . (226) We shall assume that f0 ≥ 0 and Z 1 ∞ 6 f0 ∈ L ∩ L (R ), f0 (x, v){1 + |v|2 + |x|2 + | log f0 (x, v)|}dxdv < +∞. R6

Note that the L∞ bound is assumed on the initial data. Let us recall that, in general, the Boltzmann equation may not be well defined in this space. However, here we will not consider the existence of solutions, see for some cases Section 13. That is, we shall assume that there exists a solution f ≥ 0 of the Boltzmann equation (226) such that f ∈ L∞ (]0, T [×R3x × R3v ) and f ∈ L2 (]0, T [×R3x ; L11 ∩ L log L(R3v )).

(227)

Again the boundedness of such solution f in these function spaces is a strong assumption, and it is still an open problem whether this kind of solutions exist. The definition of weak solutions uses standard formulation by the duality of the collision operator. Though it is not included in this definition, we shall also assume that we may control the entropy dissipation rate, that is D(f ) < +∞, where D(f ) ≡

Z

0

T

Z

dtdxdv

R6

Z

R3∗

Z

S2

b(f∗′ f ′ − f f∗ ) log

(228) f ′ f∗′ dv∗ dσ. f f∗

In fact, note that this assumption is about a very natural a priori bound on weak solutions. These two assumptions are not enough for obtaining the regularity estimate rigorously. We shall need a more precise lower bound: for any R > 0, inf

0≤t≤T,|x|≤R

||f (t, x, .)||L1 (R3v ) > 0.

(229)

70


Note that this assumption is on the macroscopic density associated with f . Thus, it is accordance with most works on fluid mechanics, where it is assumed that the solution is away from the vacuum. Then the partial regularity of solutions to the nonlinear Boltzmann equation is given as follows. Theorem 14.9. Partial Regularity for Nonlinear Boltzmann Equation Let f be a weak solution to the nonlinear spatially inhomogeneous Boltzmann equation (226). Let the assumptions (227), (228) and (229) hold. Then, it follows that 2ν(1−ν/2)

f ∈ Hloc 2+ν

(R7 ).

Note that the above result only gives some partial regularity and is to be compared with the result displayed in Section 13. Finally, as a last application of the uncertainty principle, one can also consider the linear Landau equation, see [13] again. 15. Spatially inhomogeneous Boltzmann equation: Hypoellipticity. We shall now present the results obtained in [14] on the Boltzmann equation (1) in dimension 3, x ∈ R3 , v ∈ R3 . Under a mild regularity assumption on the initial data, the existence of solutions and their C ∞ regularity with respect to all (time, space and velocity) variables is proven in [14]. Even though it is still not known whether only some natural bounds, such as total mass, energy and entropy on the initial data, can lead to the C ∞ regularizing effect, the results in [14] justify the C ∞ regularizing effect for the nonlinear and spatially inhomogeneous Boltzmann equation without Grad’s angular cutoff assumption. We assume that

v − v∗ π B(|v − v∗ |, cos θ) = Φ(|v − v∗ |)b(cos θ), cos θ = ,σ , 0≤θ≤ , |v − v∗ | 2 and furthermore, we suppose that the kinetic factor Φ in the cross-section is modified as γ Φ(|v − v∗ |) = 1 + |v − v∗ |2 2 , γ ∈ R. (230) As already mentioned previously, some work remains to be done to remove the smoothing assumption close to null relative velocy. Finally, the angular factor is assumed to have the following singularity. sin θ b(cos θ) ≈ Kθ−1−ν , when θ → 0+,

(231)

where 0 < ν < 2 and K is a positive constant. The following standard weighted (with respect to the velocity variable v ∈ R3 ) Sobolev spaces will be used. For m, l ∈ R, set R7 = Rt × R3x × R3v and n o ′ Hlm (R7 ) = f ∈ S (R7 ); Wl (v) f ∈ H m (R7 ) ,

which is a Hilbert space. Here H m is the usual Sobolev space, and Wl is the usual l weight Wl (v) =< v > 2 . We will also use the function spaces Hlk (R6x, v ) and Hlk (R3v ) when the variables are specified where the weight is always with respect to v ∈ R3 . Since the regularity property to be proved here is local in space and time, for convenience, we define the following local version of weighted Sobolev space. For

BOLTZMANN EQUATION

71

−∞ ≤ T1 < T2 ≤ +∞, and any given open domain Ω ⊂ R3x , define n ′ Hlm (]T1 , T2 [×Ω × R3v ) = f ∈ D (]T1 , T2 [×Ω × R3v );

o ϕ(t)ψ(x)f ∈ Hlm (R7 ) , ∀ ϕ ∈ C0∞ (]T1 , T2 [), ψ ∈ C0∞ (Ω) .

The first main result from [14] giving the smoothing effect on the solution can be stated as follows. Theorem 15.1. (Regularizing effect on solutions) Assume that 0 < ν < 2, γ ∈ R, −∞ ≤ T1 < T2 ≤ +∞ and let Ω ⊂ R3x be an open domain. Let f be a non-negative function belonging to Hl5 (]T1 , T2 [×Ω × R3v ) for all l ∈ N and solving the Boltzmann equation (1)) in the domain ]T1 , T2 [×Ω × R3v in the classical sense. Furthermore, if f satisfies the non-vacuum condition kf (t, x, ·)kL1 (R3v ) > 0,

(232)

for all (t, x) ∈]T1 , T2 [×Ω, then we have for any l ∈ N, and hence

f ∈ Hl+∞ (]T1 , T2 [×Ω × R3v ),

f ∈ C ∞ (]T1 , T2 [×Ω; S(R3v )).

With this theorem, a natural question is whether the Boltzmann equation has solution satisfying the assumptions imposed above. Let us recall the existence of both renormalized solutions and solutions bounded by two travelling Maxwellians were proved. However, neither of these results works for our purpose because of the lack of the weighted regularity Hl5 . Of course, we want to avoid the result of [121] for Gevrey class solutions! Thus, the second main result of [14] is about the local existence and uniqueness of solution for the Cauchy problem of the non-cutoff Boltzmann equation. We consider the solution in the function space with Maxwellian type exponential decay in the velocity variable. Precisely, for m ∈ R, set n o 2 E0m (R6 ) = g ∈ D′ (R6x,v ); ∃ ρ0 > 0 s.t. eρ0 g ∈ H m (R6x,v ) ,

and for T > 0

n E m ([0, T ] × R6x,v ) = f ∈ C 0 ([0, T ]; D′ (R6x,v )); o 2 ∃ ρ > 0 s.t. eρhvi f ∈ C 0 ([0, T ]; H m (R6x,v )) .

Theorem 15.2. Assume that 0 < ν < 1 and γ+ν < 1. Let f0 ≥ 0 and f0 ∈ E0k0 (R6 ) for some 4 ≤ k0 ∈ N. Then there exists T∗ > 0 such that the Cauchy problem ft + v · ∇x f = Q(f, f ), (233) f |t=0 = f0 ,

admits a non-negative and unique solution in the function space E k0 ([0, T∗ ] × R6 ). Precisely, there exists ρ > 0 such that 2

eρhvi f ∈ C 0 ([0, T∗ ]; H k0 (R6 )).

Furthermore, if we assume that the initial data f0 is in E05 (R6 ) and does not vanish on a compact set K ⊂ R3x , that is, kf0 (x, ·)kL1 (R3v ) > 0, ∀ x ∈ K,

72


then we have the regularizing effect on the above solution, that is, there exist 0 < T˜0 ≤ T∗ and a neighborhood V0 of K in R3x such that f ∈ C ∞ (]0, T˜0 [×V0 ; S(R3v )). Moreover, if γ ≤ 0, the non-negative solution of the Cauchy problem (233) is unique in the function space C 0 ([0, T∗ ]; Hpm (R6 )) for m > 3/2 + ν, p > 3/2 + 2ν. Before giving the main lines of the proofs, let us mention that it is not clear whether we can relax the regularity assumption initially made on the solutions. Note that for example, the condition that f ∈ L1 ∩ L∞ (R7 ) is enough to give a meaningful sense to a weak formulation for the spatially inhomogeneous Boltzmann equation. However, the analysis used in [14] can not be applied to this case, and so further study is needed. On the other hand, the above two theorems give an answer to a long lasting conjecture on the regularizing effect of the non-cutoff cross-sections for the spatially inhomogeneous Boltzmann equation. Let us now try to give the main lines of the proof. 15.1. Technical tools. First of all, using arguments similar to those sketched early, see for example Section 7, one may prove the following upper bound estimate on the bilinear Boltzmann collision operator (which can be extended to any velocity space dimension) Theorem 15.3. Let 0 < ν < 2 and γ ∈ R. Then for any m, α ∈ R, there exists C > 0 such that kQ(f, g)kHαm (R3v ) ≤ Ckf kL1 + α

(R3v ) +(γ+ν)+

kgkH m+ν

(234)

(R3v ) (α+γ+ν)+

m+ν 3 for all f ∈ L1α+ +(γ+ν)+ (R3v ) and g ∈ H(α+γ+ν) + (Rv ) .

We have seen that commutators with the Boltzmann operator play an important role, see in the spatially homogeneous case Sections 6 and 8. For the spatially inhomogeneous case, commutators with weights are also important, and in particular in the case of smoothed hard potentials γ > 0. One has the following results Lemma 15.4. Let l ∈ N, m ∈ R. (1) If 0 < ν < 1, there exists C > 0 such that Wl Q(f, g) − Q(f, Wl g) , h ≤ Ckf kL1 (R3 ) kgkL2 (R3 ) khkL2 (R3 ) . v v v 2 3 l+γ + l+γ + L (Rv ) (235) Moreover, if l ≥ 3 (actually, we need only l > 23 + ν), then Wl Q(f, g) − Q(f, Wl g) , h ≤ Ckf kL2 (R3 )) kgkL2 (R3 ) khkL2 (R3 ) . v + + 2 3 L (R )

l+γ

l+γ

(236) (2) If 1 < ν < 2, then for any ε > 0, there is a constant Cε > 0 such that Wl Q(f, g) − Q(f, Wl g) , h (237) 2 3 L (R ) v

≤ Cε kf kL1

(R3v ) l+ν−1+γ +

kgkH ν−1+ε

l+ν−1+γ +

(R3v ) khkL2 (R3v )

,

BOLTZMANN EQUATION

and

73

Wl Q(f, g) − Q(f, Wl g) , h 2 3 L (R )

(238)

v

≤ Cε kf kL1

l+ν−1+γ +

(R3v ) kgkL2

l+ν−1+γ +

(R3v ) khkHlν−1+ε (R3v ) .

(3) When ν = 1, we have the same estimates as (2) with ν − 1 replaced by any small κ > 0. With Lemma 15.4, we immediately have the following improved upper bound estimate of Theorem 15.3 with respect to the weight. Corollary 15.5. (1) When 0 < ν < 1, we have kQ(f, g)kHlm (R3v ) ≤ Ckf kL1

(R3v ) max{l+γ + , (γ+ν)+ }

kgkH m+ν

(R3v ) l+(ν+γ)+

,

provided that m ≤ 0 and 0 ≤ m + ν. (2) When 1 < ν < 2, we have kQ(f, g)kHlm (R3v ) ≤ Ckf kL1

(R3v ) max{l+ν−1+γ + , (ν+γ)+ }

kgkH m+ν

l+max{ν−1+γ + , (ν+γ)+ }

(R3v ) ,

(239) provided that −1 < m ≤ 0. (3) When ν = 1, we have the same form of estimate as (239) with ν − 1 replaced by any small κ > 0. For proving hypoelliptic effects, it should now be clear that we have to look for coercive type estimates. In our case, since we have to deal with weighted estimates, we shall need the following result, which is an immediate adaptation of the coercivity results mentioned in Section 5 Theorem 15.6. Assume that γ ∈ R, 0T< ν < 2. Let g ≥ 0, 6≡ 0, g ∈ L1max{γ + , 2−γ + } L log L(R3v ). Then there exists a constant Cg > 0 depending only on B(v − v∗ , θ), kgkL1 and kgkL log L , and C > + + max{γ

, 2−γ

}

1 0 depending on B(v − v∗ , θ) such that for any smooth function f ∈ Hγ/2 (R3v ) ∩ L2γ +/2 (R3v ), we have 2 − Q(g, f ), f 2 3 ≥ Cg kWγ/2 f k2H ν/2 (R3 ) − C||g||L1 3 kf k 2 L (R3 ) . + + (Rv ) v

L (Rv )

max{γ

, 2−γ

}

γ + /2

v

(240)

Let us mention that at this point, the smoothed assumption is needed, since up to now, one needs an improved version of the Coercivity results from Section 5. Furthermore, the constant Cg is seen to be an increasing function of k˜ gkL1 , k˜ gk−1 L1 1

and k˜ gk−1 ˜ =< v >−|γ| g. If the function g depends continuously on L log L where g a parameter τ ∈ Ξ, then the constant Cg depends on inf τ ∈Ξ k < v >−|γ| gτ kL1 , supτ ∈Ξ kgτ kL log L and supτ ∈Ξ kgkL1 . In the later application, we take + + max{γ

, 2−γ

}

τ = (t, x). The following interpolation inequality between weighted Sobolev spaces in v is also used, and is also a typical tool for the type of questions we have in mind, see for instance [54, 56, 39].

Lemma 15.7. For any k ∈ R, p ∈ R+ , δ > 0,

kf k2Hpk (R3v ) ≤ Cδ kf kH k−δ (R3 ) kf kH k+δ (R3 ) . 2p

v

0

v

(241)

74


Finally, as a last technical tool, we need to study the commutators of a family of pseudo-differential operators with the Boltzmann collision operator. This is a key step in the regularity analysis of weak solutions because it requires the mollifiers defined by pseudo-differential operators. Proposition 15.8. Let λ ∈ R and M (ξ) be a positive symbol of pseudo-differential λ ˜ (|ξ|2 ). Assume that for any c > 0 there operator in S1,0 of the form of M (ξ) = M exists a constant C > 0 such that for any s, τ > 0 ˜ (s) s M c−1 ≤ ≤ c implies C −1 ≤ ≤ C. (242) ˜ (τ ) τ M Furthermore assume that M (ξ) satisfies |M (α) (ξ)| = |∂ξα M (ξ)| ≤ Cα M (ξ) < ξ >−|α| ,

for any α ∈ N3 . Then the followings hold. (1) If 0 < ν < 1, for any N > 0 there exists a CN > 0 such that (M (Dv )Q(f, g) − Q(f, M (Dv )g), h)L2 (R3 ) v ≤ CN kf kL1 + (R3v )) kM gkL2 + (R3v ) + kgkH λ−N (R3 )) khkL2(R3v ) . γ

γ+

γ

(243)

(244)

v

(2) If 1 < ν < 2, for any N > 0 and any ε > 0 there exists a CN,ε > 0 such that (M (Dv )Q(f, g) − Q(f, M (Dv )g), h)L2 (R3 ) (245) v ≤ CN,ε kf kL1 kM gkH ν−1+ε (R3 )) + kgkH λ−N (R3 )) khkL2(R3v ) . 3 + (Rv )) (ν+γ−1)

(ν+γ−1)+

v

γ+

v

(3) If ν = 1, we have the same estimate as (245) with (ν + γ − 1) replaced by (γ + κ) for any small κ > 0.

15.2. Regularizing effect. With the previous technical tools, we will now sketch the proof of the regularizing effect on solutions to the non-cutoff Boltzmann equation starting from f ∈ Hl5 (]T1 , T2 [ ×Ω × R3v )). Actually this is proven by using an induction argument: 1) in the first step, one shows the gain of regularity in the variable v mainly by using the singularity in the cross-section, that is, the coercivity property; 2) in a second step, we apply the hypo-elliptic estimate obtained by a generalized version of the uncertainty principle to show the gain of regularity in (x, t) variables, from Section 14; 3) then an induction argument will lead to at least one order higher regularity in (x, t) variables; 4) by using the equation and an induction argument again, at least one order higher regularity can be obtained in v variable. Therefore, the solution is shown to be in Hl6 (]T1 , T2 [ ×Ω × R3v ) which by induction leads to Hl∞ (]T1 , T2 [ ×Ω × R3v ). We now start the proof. Let f ∈ Hl5 (]T1 , T2 [ ×Ω × R3v )), for all l ∈ N, be a (classical) solution of the Boltzmann equation (1). We now want to prove the full regularity of ϕ(t)ψ(x)f for any smooth cutoff functions ϕ ∈ C0∞ (]T1 , T2 [), ψ ∈ C0∞ (Ω). 1) Initialization Here and below, φ denotes a cutoff function satisfying φ ∈ C0∞ and 0 ≤ φ ≤ 1. Notation φ1 ⊂⊂ φ2 stands for two cutoff functions such that φ2 = 1 on the support of φ1 .

BOLTZMANN EQUATION

75

Take the smooth cutoff functions ϕ, ϕ2 , ϕ3 ∈ C0∞ (]T1 , T2 [) and ψ, ψ2 , ψ3 ∈ such that ϕ ⊂⊂ ϕ2 ⊂⊂ ϕ3 and ψ ⊂⊂ ψ2 ⊂⊂ ψ3 . Set f1 = ϕ(t)ψ(x)f , f2 = ϕ2 (t)ψ2 (x)f and f3 = ϕ3 (t)ψ3 (x)f . For α ∈ N7 , |α| ≤ 5, define C0∞ (Ω)

α g = ∂ α (ϕ(t)ψ(x)f ) = ∂t,x,v (ϕ(t)ψ(x)f ) ∈ L2l (R7 ).

Firstly, the translation invariance of the collision operator w.r.t. the variable v implies that (see [56, 121] ), for the translation operation τh in v by h, we have τh G(f, g) = Q(τh f, τh g). Then the Leibniz formula with respect to the t, x variables yields the following equation in a weak sense gt + v · ∂x g = Q(f2 , g) + G, (t, x, v) ∈ R7 , where G

=

X

α1 +α2 =α, 1≤|α1 |

Cαα21 Q ∂ α1 f2 , ∂ α2 f1

(246)

(247)

+ ∂ α ϕt ψ(x)f + v · ψx (x)ϕ(t)f + [∂ α , v · ∂x ](ϕ(t)ψ(x)f ) ≡ (A) + (B) + (C).

To prove the regularity of g = ∂ α (ϕ(t)ψ(x)f ), the natural idea would be to use g as a test function for equation (246). But at this point, g only belongs to L2l (R7 ) so that it is only a weak solution to equation (246). By using the upper bound estimate on Q, we have Q(f2 , g) ∈ L2 (R4t,x ; H −2s (R3v )). Thus, we need to choose the test functions at least in the space L2 (R4t,x ; H 2s (R3v )). For this, we will use a mollification of g with respect to the variables (x, v) as a test function. For this purpose, let S ∈ C0∞ (R) satisfy 0 ≤ S ≤ 1 and S(τ ) = 1, |τ | ≤ 1;

S(τ ) = 0, |τ | ≥ 2.

Then SN (Dx )SN (Dv ) = S(2−2N |Dx |2 )S(2−2N |Dv |2 ) : Hl−∞ (R6 ) → Hl+∞ (R6 ), is a regularization operator such that k SN (Dx )SN (Dv )f − f kL2l (R6 ) → 0,

as N → ∞.

Choose another cutoff function ψ ⊂⊂ ψ1 ⊂⊂ ψ2 and set PN, l = ψ1 (x)SN (Dx ) Wl SN (Dv ).

Then we can take ⋆ 1 +∞ g˜ = PN, (R6 )) l (PN, l g) ∈ C (R; H

as a test function for the equation (246). It follows by integration by parts on R7 = R1t × R3x × R3v that [SN (Dv ), v] · ∇x SN (Dx )g, ψ1 (x)Wl PN,l g = L2 (R7 ) PN,l Q(f2 , g), PN,l g + G, g˜ , L2 (R7 )

L2 (R7 )

76


which implies that − Q(f2 , PN, l g), PN, l g 2 7 L (R ) = − [SN (Dv ), v] · ∇x SN (Dx )g, ψ1 (x)Wl PN,l g 2 7 L (R ) + PN, l Q(f2 , g) − Q(f2 , PN, l g), PN, l g + G, g˜ L2 (R7 )

(248)

L2 (R7 )

.

By using (248), we can deduce the regularity of g from the coercivity property of the collision operator on the left hand side and the upper bound estimate on the right hand side, using the technical tools from the previous sub-section. 2) Gain of regularity in v By using coercivity from the technical tools, a partial smoothing effect of the cross-section on the weak solution g in the velocity variable v may be proven Proposition 15.9. Assume that 0 < ν < 2, γ ∈ R. Let f ∈ Hl5 (]T1 , T2 [ ×Ω × R3v ) be a solution of the equation (1) for all l ∈ N. Assume furthermore that f (t, x, v) ≥ 0 and kf (t, x, ·)kL1 (R3v ) > 0,

(249)

for all (t, x, v) ∈]T1 , T2 [ ×Ω × R3v . Then one has, 5 7 Λν/2 v f1 ∈ Hl (R ),

(250)

for any l ∈ N, where f1 = ϕ(t)ψ(x)f with ϕ ∈ C0∞ (]T1 , T2 [), ψ ∈ C ∞ (Ω). 3) Commutators tools We now turn to the estimates of commutators between the mollification operators and the collision operator, which are given in the following two lemmas. Lemma 15.10. For any γ ∈ R, we have (1) If 0 < ν < 1, then for any suitable functions f and g with the following norms well defined, one has kSN (Dv )Q(f, g) − Q(f, SN (Dv )g)kL2 (R3v ) ≤ Ckf kL1 + (R3v ) kgkL2 + (R3v ) , γ

γ

(251)

for some constant C independent of N . (2) If 1 < ν < 2, then for any δ > 0 there exists a constant Cδ > 0 such that kSN (Dv )Q(f, g) − Q(f, SN (Dv )g)kL2 (R3v ) ≤ Cδ kf kL1

(R3v ) (ν+γ−1)+

kgkH ν−1+δ

(R3v ) (ν+γ−1)+

,

(252)

and kSN (Dv )Q(f, g) − Q(f, SN (Dv )g)kH 1−ν−δ (R3 ) ≤ Cδ kf kL1

(R3v ) (ν+γ−1)+

kgkL2

(R3v ) (ν+γ−1)+

(253)

.

(3) When ν = 1, we have the same form of estimate as (252) with (ν + γ − 1) replaced by (γ + κ) for any small κ > 0. The following lemma is on the commutator of the collision operator with mollifier in the x variable.

BOLTZMANN EQUATION

77

Lemma 15.11. Let 0 < ν < 2 and γ, m ∈ R. For any suitable functions f and h with the following norms well defined, one has kSN (Dx )Q(f, h) − Q(f, SN (Dx ) h)kL2 (R4t,x , H m−ν (R3v )) ≤ C2−N k∇x f kL∞ (R4t,x , L1

(R3v )) (ν+γ)+

(254)

khkL2 (R4t,x , H m

(R3v )) (ν+γ)+

.

for a constant C independent of N . In particular, if we apply (254) with h = SN (Dv )g and m = 1, we get kSN (Dx )Q(f, SN (Dv )g) − Q(f, SN (Dx )SN (Dv )g)kL2 (R4t,x , H 1−ν (R3v ))(255) ≤ Ck∇x f kL∞ (R4t,x , L1

(R3v )) (ν+γ)+

kgkL2 (R4t,x , L2

(R3v )) (ν+γ)+

.

Here, we have used the fact that a mollification operator SN (Dv ) in the v variable has the property that k2−N SN (Dv )g(t, x, · )kH 1

(R3v ) (ν+γ)+

≤ Ckg(t, x, · )kL2

(R3v ) (ν+γ)+

,

where C is a constant independent on N . 4) Gain of regularity in (t, x) We now continue with the application of the uncertainty principle from Section 14. For this purpose, the following Leibniz type formula for fractional derivatives with respect to variable (t, x) is needed Lemma 15.12. Let 0 < λ < 1. Then there exists a positive constant Cλ 6= 0 such that for any f ∈ S(Rn ), one has Z f (y) − f (y + h) λ −1 λˆ dh. (256) |Dy | f (y) = F |ξ| f (ξ) = Cλ |h|n+λ n R Using this Lemma, we have the following Leibniz type formula, Z f (y)g(y) − f (y + h)g(y + h) λ |Dy | f (y)g(y) = Cλ dh |h|n+λ n R = +

g(y)|Dy |λ f (y) + f (y)|Dy |λ g(y) Z f (y) − f (y + h) g(y + h) − g(y) Cλ dh. |h|n+λ Rn

(257) (258)

We now turn to the analysis of the fractional derivative with respect to (t, x) of the nonlinear collision operator. Denote the difference with respect to (t, x) by fh (t, x, v) = f (t, x, v) − f ((t, x) + h , v),

h ∈ R4t,x .

It follows that for the collision operator, |Dt,x |λ Q f, g = Q |Dt,x |λ f, g + Q f, |Dt,x |λ g +Cλ

Z

R4

|h|−4−λ Q fh , gh dh.

(259)

78


This kind of decomposition is used extensively in order to get the partial regularity with respect to the (t, x) variable. Then, we have the following proposition on the gain of regularity in the variable (t, x) through the uncertainty principle from Section 14. Proposition 15.13. Under the hypothesis of Theorem 15.1, one has 0 Λst,x f1 ∈ Hl5 (R7 ),

for any l ∈ N and 0 < s0 =

(260)

ν(1−ν/2) (2+ν) .

Therefore, under the hypothesis f ∈ Hl5 (]T1 , T2 [×Ω × R3v ) for all l ∈ N, it follows that for any l ∈ N we have 5 7 Λν/2 v (ϕ(t)ψ(x)f ) ∈ Hl (R ),

0 Λst,x (ϕ(t)ψ(x)f ) ∈ Hl5 (R7 ) .

(261)

This partial regularity in (t, x) variable can now be improved using the following result Proposition 15.14. Let 0 < λ < 1. Suppose that f ∈ Hl5 (]T1 , T2 [ ×Ω × R3v ) is a solution of the equation (1) for all l ∈ N. Furthermore, assume that for any cutoff functions ϕ, ψ, 5 7 Λν/2 v (ϕ(t)ψ(x)f ) ∈ Hl (R ),

Λλt,x (ϕ(t)ψ(x)f ) ∈ Hl5 (R7 ).

(262)

Then, one has λ 5 7 Λν/2 v Λt,x (ϕ(t)ψ(x)f ) ∈ Hl (R ),

(263)

for any l ∈ N and any cutoff functions ϕ, ψ.

5) Improvement of the regularity One can prove the following regularity result on the solution with respect to the (t, x) variable. Proposition 15.15. Under the hypothesis of Theorem 15.1, one has 5 7 Λ1+ε t,x (ϕ(t)ψ(x)f ) ∈ Hl (R ),

(264)

for any l ∈ N and some ε > 0.

6) Final steps of the proof of Theorem 15.1 From Propositions 15.9 and 15.15, it follows that for any l ∈ N, Λν/2 (ϕ(t)ψ(x)f ), v

∇t,x (ϕ(t)ψ(x)f ) ∈ Hl5 (R7 ).

(265)

These facts are used to get the high order regularity with respect to the variable v. Proposition 15.16. Let 0 < λ < 1. Suppose that, for any cutoff functions ϕ ∈ C0∞ (]T1 , T2 [), ψ ∈ C0∞ (Ω) and all l ∈ N, Λλv (ϕ(t)ψ(x)f ),

∇x (ϕ(t)ψ(x)f ) ∈ Hl5 (R7 ).

(266)

Then, for any cutoff function and any l ∈ N,

Λλ+ν/2 (ϕ(t)ψ(x)f ) ∈ Hl5 (R7 ). v

(267)

We can now conclude the following regularity result with respect to the variable v, which is proven by induction.

BOLTZMANN EQUATION

79

Proposition 15.17. Under the hypothesis of Theorem 15.1, one has 5 7 Λ1+ε v (ϕ(t)ψ(x)f ) ∈ Hl (R ),

(268)

for any l ∈ N and some ε > 0.

Finally, from Proposition 15.15 (more precisely (264)) and Proposition 15.17, we can now deduce that, for any l ∈ N, and any cutoff functions ϕ(t) and ψ(x), ϕ(t)ψ(x)f ∈ Hl6 (R7 ).

The proof of Theorem 15.1 is then completed by induction. Indeed, if f is a solution of Boltzmann equation satisfying the assumptions of Theorem 15.1, then, when m ≥ 5, we have f ∈ Hlm (]T1 , T2 [×Ω × R3v ), ∀l ∈ N =⇒ f ∈ Hlm+1 (]T1 , T2 [×Ω × R3v ), ∀l ∈ N.

Thus, the full regularity of Theorem 15.1 is obtained by induction from m = 5.

15.3. Existence and uniqueness of local solutions. The local existence of solutions to the spatially inhomogeneous Boltzmann equation without angular cutoff is so far not well studied. The strategy of proving the existence result given by Theorem 15.2 is to approximate the non-cutoff cross-section by a family of cutoff cross-sections and approximate the Boltzmann equation by a sequence of iterative linear equations. Then by proving the existence of these approximate linear equations and by obtaining a uniform estimate on the solutions with respect to the cutoff parameter in some suitable weighted Sobolev space, the compactness will lead to the convergence of the approximate solutions to the desired solution for the original problem. One of the techniques used here is to introduce a transformation defined by the time dependent Maxwellian developed in [121]. The purpose of this transformation is to get an extra gain of one order higher weight in the velocity variable at the expense of the loss of the decay in the time dependent Maxwellian. Moreover, the uniqueness of the solution is also proved in some function space. We refer for details again to [14]. 16. Spatially homogeneous Boltzmann equation: Other results. First of all, the search for Lp and moments estimates is quite well understood for Boltzmann spatially homogeneous equation, in the cutoff case. It is therefore natural to study this question for singular kernels. Of course, the spatially homogeneous case is still always simpler to consider. Other important questions are related to stability, uniqueness results. 16.1. Weighted Lp estimates. We first wish to show some weighted Lp (moments) estimates for solutions of the spatially homogeneous Boltzmann equation, and for singular kernels. First results in this direction are due to Desvillettes and Mouhot [51]. However, we shall sketch a method of proof using some ideas already introduced in the context of renormalisation for the spatially inhomogeneous Boltzmann equation. As such, it is then possible to consider more general nonlinearities. But firstly, we state informally the results of Desvillettes and Mouhot [51] Theorem 16.1. Under suitable assumptions on the singular kernel, on p (integrability index) and q (weight index), assume that f0 belongs to L2max(p,2)q+2 ∩ Lpq . Then, one has: 1) there exists a weak solution of the Boltzmann equation in the L∞ (0, +∞, Lpq ). 2) if γ > 0, then this solution belongs to L∞ (τ, +∞; Lpr ), for all τ > 0 and τ > q.

80


We shall present a different proof from [51], but some calculations are similar. The main difference is that we shall use essentially Taylor formula at order one for (nonlinear) functions β : R+ → R, in the following form (assuming smoothness): for all a, b in R+ , β ′ denoting the derivative of β β ′ (a).(b − a) = [β(b) − β(a)] − Rβ (a, b)

(269)

where Rβ (a, b) denotes the reminder in Taylor formula; so it is defined alternatively as Rβ (a, b) = [β(b) − β(a)] − β ′ (a).(b − a) (270) or also by Z (b − a)2 1 Rβ (a, b) = (1 − u)β”(a + u(b − a))du. (271) 2 0 Of particular interest is the case when β is convex on R+ , thus when Rβ (a, b) ≥ 0; in particular, this the case for the choice β(s) = sp , p ≥ 1, made by Desvillettes and Mouhot. Let a positive function f = f (t, v) ≥ 0, t ∈ R+ time variable, v ∈ Rn velocity variable, be solution in standard sense of spatially homogeneous Boltzmann equation: ∂t f (t, v) = Q(f, f )(t, v), (272) where Q is Boltzmann collisional operator. Recall that we use the notation, for smooth functions g and f , Z Z Q(g, f )(v) = dv∗ dσB(|v − v∗ |, cos θ)[g∗′ f ′ − g∗ f ]. (273) Rn

S n−1

Function B in (273), in general, may be assumed to satisfy the extremely large singular assumptions as in [9], [16], and as we have recalled in previous Sections. However, only for simplifications, as may be checked from the proofs, it is enough to have in mind the following special case: B(v − v∗ , cos θ) = |v − v∗ |γ b(cos θ).

(274)

Above γ > 0, b(cos θ) is bounded away from θ = 0 and has a singularity for θ ∼ 0 as follows c sinn−2 θb(cos θ) ∼ 1+ν as θ → 0, (275) θ for some fixed non negative constants c and 0 < ν < 2 with the following link s−5 2 and ν = , for some s > 2. γ= s−1 s−1 We shall also simply write B(., .), omitting the arguments of B. One has the following result Proposition 16.2. Under the above assumptions, choose β(u) = up , for some p ≥ 1. Then, for any positive and smooth function f , for all α ≥ 0, with pα ≥ 1, one has Z Z Z Z dvQ(f, f )β ′ (f ) < v >pα + dv dv∗ dσB(., .)g∗′ Rβ (f, f ′ ) Rn

≤C +

−

+

Rn p kf kLpα −

Rn

K

−

S n−1 p kf kLp γ . α+

p

Above, C and K are positive constants depending on an upper bound of kf kL1pα+2 , on an upper bound of the entropy and on a lower bound of kf kL1 .

BOLTZMANN EQUATION

81

This result is only but a slight improvement of Proposition 2.5 of [51]. Of course, from there, one can deduce the main Theorem 1.1 of [51], by using the improved integrability for the gain term, as shown by Mouhot and Villani [112], following previous works of Wennberg [140], simplifying the first proof by Lions [91]. Let us note that it is still an open question to get this improved integrability directly. The (slight) improvements w.r.t. [51] of the proof below are: - it is possible to consider more general nonlinearities β; - we are also able to get a term similar to the entropy dissipation rate; this is the second term on the l.h.s of the above inequality, see [9] for possible implications. The rest of this sub-section is devoted to the short proof of the above Propositon. Moreover, from time to time, g denotes any smooth positive function, that we shall set to be f , but only in the final steps. By this way, it should be certainly possible to have estimations, but in a linear context. We first use ideas from renormalised solutions, in the context of singular kernels, see [16, 3]. Let φ = φ(v) be a smooth function. In the applications we have in mind, it will be taken as φ(v) =< v >pα , for p ≥ 1, α ∈ R+ . Let β a smooth function, β ′ its derivative, and Rβ its remainder. Then, < .; . > denoting the standard duality bracket, one has Z Z Z o n < Q(g, f ); φβ(f ) >= dv dv∗ dσB(., .) g∗′ (f ′ − f ) + f (g∗′ − g∗ ) φβ ′ (f ). Rn

S n−1

Rn

′

Beware of the fact that β is the derivative and f ′ is the value of f at v ′ ! Splitting the term, we get < Q(g, f ); φβ(f ) >= A1 + A2 , where A1 ≡ A2 ≡

Z

Z

Z Z dv dv∗

Rn

dσB(., .)g∗′ (f ′ − f )φβ ′ (f ),

(276)

dσB(., .)(g∗′ − g∗ ).

(277)

S n−1

Rn

′

dvφf β (f )

Rn

Z

Z

dv∗

Rn

S n−1

The treatment of A2 from (277) should now be clear in view of [9], [16]. That is, using the Cancellation Lemma therein, see Section 4, one has Z A2 = dvφf β ′ (f ).S ∗ g(v), Rn

where the convolution kernel S is given by the rule in the above mentioned papers. Thus, one can get upper bounds estimates on it. For A1 given by (276), we now use the convexity of β to get Z Z Z A1 ≤ dv dv∗ dσB(., .)g∗′ φ[β(f ′ ) − β(f ) ≡ B. (278) Rn

Rn

S n−1

Note that Dβ , the term we have missed, can be (having in mind Boltzmann equation) put on the l.h.s. of the equation, which also gives an information if for example β was superquadratic. Of course, we have defined Dβ ≥ 0 by the formula A1 = B − Dβ , that is also Z Z Z dv dv∗ dσB(., .)g∗′ Rβ (f, f ′ ). (279) Dβ ≡ Rn

Rn

S n−1

82


Then one can write B given by (278) as follows Z Z Z B= dv dv∗ dσB(., .)g∗′ [β(f ′ )φ′ − β(f )φ] Rn Rn S n−1 Z Z Z + dv dv∗ dσB(., .)g∗′ n n n−1 R R S Z Z Z = B(., .)g∗′ β(f ′ )φ′ − g∗ φβ(f ) Z Z Z Z Z Z − φβ(f ) (g∗′ − g∗ ) + g∗′ β(f ′ )(φ′ − φ).

We note that the first term is 0 since this is nothing else than the integral of Q(g, φβ(f )). By gluing all the above estimates, we have obtained:

=

Z

< Q(g, f ); φβ ′ (f ) > +Dβ dvφ[f β ′ (f ) − β(f )].S ∗ g(v)+ < Q(g, β(f )); φ >≡ B1 + B2 .

(280)

Rn

Note also that B2 ≡< Q(g, β(f )); φ >=

Z

Z Z dv dv∗

Rn

Rn

dσB(., .)g∗ β(f )[φ′ − φ].

S n−1

From [9] and also its proof, we know that, for a.a. v ∈ Rn , Z Sg ≡ dv∗ dσ B(v − v∗ , σ)(g∗′ − g∗ ) = g ∗v S,

(281)

Rn ×S n−1

where

|z| S(|z|) = |S | dθ sin , cos θ − B(|z|, cos θ). . cos(θ/2) 0 (282) Making the explicit choice of the special assumption (though it is not really necessary), we obtain S(|z|) = cb |z|γ , and thus Z Sg(v) = cb dv∗ |v − v∗ |γ g(v∗ ) n−2

Z

π 2

n−2

1 θ B cosn (θ/2)

Rn

and

|Sg(v)| ≤ cb < v >γ kgkL1γ .

Making now the choice φ =< v >pα and β(s) = sp , for p ≥ 1, we obtain Z B1 ≡ φ[f β ′ (f ) − β(f )].S ∗ g(v) ≤ CkgkL1γ kf kpLp γ . α+

p

It remains to deal with the last term B2 and for this purpose, we need to have an estimate for Z γ E = |v − v∗ | b(cos θ)[φ′ − φ], S n−1

where recall that φ =< v >pα ; this fails under the scope of previous papers, and in particular [16], but we can proceed differently here, assuming for simplicity that 0 < ν < 1. First note that | < v ′ >pα − < v >pα | .< v >pα + < v∗ >pα . Furthermore, by using Taylor formula at order one (this is enough in the case 0 < ν < 1; in the case ν ≥ 1, one needs Taylor’s formula at order two, the symetrization used in [16] and similar computations as done below), one obtains

BOLTZMANN EQUATION

83

θ | < v ′ >pα − < v >pα | . |v − v∗ | sin [< v >pα−1 + < v∗ >pα−1 ]. 2 Set M1 = |v − v∗ |[< v >pα−1 + < v∗ >pα−1 ] and M2 =< v >pα + < v∗ >pα (up to unimportant constants). We have just shown that θ | < v ′ >pα − < v >pα | . min{M1 sin , M2 }. 2 Computing a spherical integral, one gets E . |v − v∗ |γ+ν {< v >pα + < v∗ >pα }−ν+1 {< v >pα−1 + < v∗ >pα−1 }ν .< v >pα+γ < v∗ >γ+ν + < v >γ+ν < v∗ >pα+γ +

+ < v >pαν+γ < v∗ >pα(−ν+1)+γ+ν + < v∗ >pαν+γ < v >pα(−ν+1)+γ+ν . Thus we have obtained  p p  B2 ≡< Q(g, β(f ); φ >. kgkL1γ+ν kf kLp γ + kgkL1pα+γ kf kLp α+

 

+kgkL1pα(−ν+1)+γ+ν kf kpLp

γ αν+ p

+

p

kgkL1pαν+γ kf kpLp

γ+ν p

γ+ν α(−ν+1)+ p

.

(283)

For the usual inverse power law potentials, one has for some s > 2 s−5 2 γ= and ν = . s−1 s−1

In particular γ + ν = s−3 s−1 , −1 < γ + ν < 1. So if we choose p and α such that pα is bigger than 1, then we get directly Corollary 2.2 from [51]. Next, the point is that if we consider singular kernels, but cutoff for large values of θ, that is with factor Isin θ ≤δ , then the same result holds, up to a factor of δ 2−ν 2 (δ 1−ν also if we assume that 0 < ν < 1). We now use ideas of renormalisation, but this time for non singular kernels, as introduced by Di Perna and Lions [58, 91]. According to the last part of the previous section, let us fix δ > 0 and denote by Bδ the kernel truncated at level δ; that is multiplied by Isin θ ≥δ . We denote by Qδ 2 the corresponding collision operator; then it follows that, with usual notations: − ′ ′ < Qδ (g, f ); φβ ′ (f ) >=< Q+ δ (g, f ); φβ (f ) > − < Qδ (g, f ); φβ (f ) > .

By definition, one has
=

Z

Z Z dv dv∗

Rn

Rn

dσg∗ f φ(v ′ )β ′ (f ′ )Bδ ,

S n−1

that we split into two parts according to wether or not |v| ≤ R for some large R, ′ < Q+ δ (g, f ); φβ (f ) >= IR− + IR+ ,

where

Z

Z Z IR− ≡ dv dv∗ dσg∗ f I|v|≤R φ(v ′ )β ′ (f ′ )Bδ Rn Rn S n−1 Z Z Z IR+ ≡ dv dv∗ dσg∗ f I|v|≥R φ(v ′ )β ′ (f ′ )Bδ . Rn

Rn

(284) (285)

S n−1

We first deal with IR− that we split, for some large parameter µ1 as follows Z Z Z IR− = dv dv∗ dσg∗ f I|v|≤R φ(v ′ )β ′ (f ′ )If ′ ≤µ1 f Bδ Rn

Rn

S n−1

84


Z

+

Z Z dv dv∗

Rn

Rn

dσg∗ f I|v|≤R φ(v ′ )β ′ (f ′ )If ≤ µ1

1

S n−1 p

f ′ Bδ .

With our choice of β(u) = u , it follows that Z Z Z IR− . dv dv∗ dσg∗ φ(v ′ )I|v|≤R pµp−1 β(f )Bδ 1 Rn

+

Z

Rn

.

Rn

S n−1

Z Z dv dv∗ Rn

dσg∗ φ(v ′ )I|v|≤R p

S n−1

1

β(f µp−1 1

′

)Bδ

1 1 1 γp p−1 R µ1 kgkL1pα+γ kf kpLpα + ν p−1 kgkL1γ kf kpLp γ . δν δ µ1 α+ p

The factor δ1ν comes from the explicit computation of the spherical integral of Bδ ; we have also used the Cancellation Lemma (or more precisely its proof, see [9]) to get the second term. Next, for IR+ , making the change of variables “v → v∗ ”, one gets first Z Z Z IR+ = dv dv∗ dσgf∗ I|v∗ |≥R φ(v∗′ )β ′ (f∗′ )Bδ . Rn

Rn

S n−1

Then, we decompose, for a parameter µ2 , this last integral according to whether or not g ≤ µ2 f∗′ , Z Z Z IR+ = dv dv∗ dσgf∗ I|v∗ |≥R Ig≤µ2 f∗′ φ(v∗′ )β ′ (f∗′ )Bδ Rn

+

Z

Rn

Z Z dv dv∗

Rn

Rn

S n−1

dσgf∗ I|v∗ |≥R If∗′ ≤ µ1 g φ(v∗′ )β ′ (f∗′ )Bδ . 2

S n−1

Using moreover the change of variables σ → −σ for the first integral (as in [51]), ˜ we obtain, taking now g = f , for a suitable kernel B Z Z Z ˜δ IR+ . µ2 p dv dv∗ dσf∗ I|v∗ |≥R φ(v ′ )β(f ′ )B Rn

+p . µ2 p

1 µp−1 2

Z

Rn

Z Z dv dv∗

Rn

Rn

S n−1

dσf∗ I|v∗ |≥R φ(v∗′ )β ′ (f )Bδ

S n−1

1 p 1 kf I|v|≥R kL1γ kf kpLp γ + p−1 ν kf I|v|≥R kL1pα+γ kf kpLp γ δν α+ α+ µ2 δ p p

. µ2 p

1 1 p 1 kf kL12 kf kpLp γ + p−1 ν kf kL1pα+γ kf kpLp γ . ν 2−γ δ R α+ α+ µ2 δ p p

It follows that ′ < Q+ δ (g, f ); φβ (f ) >.

+µ2 p

1 1 1 γp p−1 R µ1 kgkL1pα+γ kf kpLpα + ν p−1 kgkL1γ kf kpLp γ δν δ µ1 α+ p

1 1 p 1 kf kL12 kf kpLp γ + p−1 ν kf kL1pα+γ kf kpLp γ . ν 2−γ δ R α+ α+ µ2 δ p p

Then we note finally that (again as in [51]), p ′ − − < Q− δ (f, f ); φβ (f ) >. −K kf kLp

α+

γ p

To finish, we choose δ small enough, then µ1 large enough, then µ2 large enough and finally R large enough to get the result.

BOLTZMANN EQUATION

85

16.2. Other results: Stability, uniqueness. Let us refer to Desvillettes [45] for some open problems in the 2000’s. Let us first of all present the stability and uniqueness results from recent works by Desvillettes and Mouhot [53]. The assumptions on the collision kernel are quite similar to previously stated ones: B = Φ(|v − v∗ |)b(cos θ), (286) the angular part is nonegative, smooth (or at least locally integrable) for θ ∈ (0, π] and such that b(cos θ) ∼θ→0 Cb θ−(N −1)−ν , ν ∈ (−∞, 2). (287) For the kinetic part Φ, one of the following assumption will be supposed to hold true:   z 7→ Φ(|z|) is strictly positive, C ∞ and such that Φ(|z|) ∼|z|∼→+∞ CΦ |z|γ for some CΦ > 0 and γ ∈ (0, 1], (288)  ∀z ∈ Rn , p ∈ N∗ , |Φ(p) (|z|) ≤ CΦ,p for some CΦ,p > 0.   z 7→ Φ(|z|) is strictly positive, C ∞ , such that Φ(|z|) ∼|z|→+∞ CΦ |z|γ for some CΦ > 0 and γ ∈ (−n, 0], (289)  ∀z ∈ Rn , p ∈ N∗ , |Φ(p) (|z|) ≤ CΦ,p for some CΦ,p > 0. z 7→ Φ(|z|) is given by the explicit formula (290) Φ(|z|) = CΦ |z|γ , for some CΦ > 0 and γ ∈ (0, 1]. z 7→ Φ(|z|) is given by the explicit formula (291) Φ(|z|) = CΦ |z|γ , for some CΦ > 0 and γ ∈ (−n, 0]. The stability result of Desvillettes and Mouhot uses an important assumption of regularity on solutions. This is because their methods is based on integration by parts arguments. However, it should now be possible to get similar results without such an assumption by using the calculations that we mentionned in the previous Sections. Theorem 16.3. Let B be a collison kernel such that (286) and (287) hold true for some ν < 1. Let f , g in L∞ ([0, T ]; L12 ∩ L log L(Rn )) be two nonnegative solutions of the spatially homogeneous Boltzmann equation associated with B, on some time interval [0, T ]. 1) Under assumptions (288), (289) or (290), for any q ≥ 2, on has the following a priori bound ∀t ∈ [0, T ], ||f (t, .) − g(t, .)||L1q ≤ ||f0 − g0 ||L1q exp(Cs t)

where

Cs ≃ ( sup ||f (t, .)||W 1,1

q+(1+γ)+

t∈[0,]

+ ||g(t, .)||W 1,1

q+(1+γ)+

(292)

)

2) Under assumption (291), (292) still holds true for any q ≥ 2 but with Cs ≃ ( sup ||f (t, .)||W 1,1 t∈[0,]

q+(1+γ)+

∩Lp1

+ sup ||∇f (t, .)||Lp2 + ||g(t, .)||W 1,1 t∈[0,T ]

+ sup ||∇g(t, .)||Lp2 t∈[0,T ]

q+(1+γ)+

∩Lp1 )

86


with p1 > n/(n + γ) and p2 > n/(n + γ + 1). A similar result is also available for strong angular singularities, see [53] for further details. From this result and propagation of smoothness studies (which anyway should be carefully written), one deduces the following uniqueness result Theorem 16.4. Assume (286)-(287), for some ν < 1. Then: 1) if (288), (289) or (290) hold true, and f (0, .) belongs to Wq1,1 for some q ≥ 2, there is a unique global solution of the spatially homogeneous Boltzmann equation in the space Wq1,1 . 2) if (291) holds true for some γ ≥ −1 and f (0, .) belongs to Wq1,1 ∩ Lp for p > n/(n + γ) and q ≥ 2, then there is a unique solution on a certain time interval [0, T ] in the space Wq1,1 ∩ Lp , which is global when γ ∈ (−ν, 0] and q is big enough.

As mentioned there, the strong angular case ν > 1 is still open, by using this kind of technics. However, a recent result by Fournier and Guerin [65] solves this issue, by using a probabilistic framework, and in particular Tanaka’s old ideas (which are linked with Wasserstein distance). We now present this framework and the assumptions of [65]. We are now working in dimension 3.

Z

B sin θ = Φ(|v − v∗ |)β(θ), π 0

β(θ)dθ = +∞, κ1 ≡

Z

π

θ2 β(θ)dθ < +∞.

(293) (294)

0

Let us introduce, for θ ∈ (0, π] Z π H(θ) ≡ β(x)dx and G(z) ≡ H −1 (z).

(295)

θ

Then, we note that H is a continuus decreasing bijection from (0, π] into [0, +∞) and its inverse function G : [0, +∞) → (0, π] is defined by G(H(θ)) = θ and H(G(z)) = z. It is assumed that there exists κ2 > 0 such that for all x, y in R+ Z ∞ (x − y)2 (G(z/x) − G(z/y))2 dz ≤ κ2 . (296) x+y 0 For the velocity part of the cross section, we assume that for all x, y in R+  2 2   min(x2 , y 2 ) [Φ(x)−Φ(y)] Φ(x)+Φ(y) + (x − y) [Φ(x) + Φ(y)] (297)   min(x, y)|x − y||Φ(x) − Φ(y)| ≤ (x − y)2 [Ψ(x) + Ψ(y)], for some function Ψ : R+ → R+ with for some γ ∈ (−3, 0], some κ3 such that

Ψ(x) ≤ κ3 xγ . (298) It follows in particular that for all x ∈ R+ , one has Φ(x) ≤ Ψ(x) and then Φ(x) ≤ κ3 xγ . While (296) is very satisfyng, (298) with γ = 0 corresponds to regularized velocity cross sections, while general assumption (298) allows to deal with soft potentials, in view of the following

BOLTZMANN EQUATION

87

Lemma 16.5. (i) Assume that 0 < ν < 2 and that cθν−1 ≤ β(θ) ≤ Cθ−ν−1 . Then (294) and (296) hold true. (ii) Assume that Φ(x) = min(xα , A) for some A > 0, some α ∈ R, or that Φ(x) = (ε + x)α , for some ε > 0, α < 0. Then (298) with γ = 0 holds true. (iii) Assume that for some γ ∈ (−3, 0], Φ(x) = xγ . Then (298) with this value of γ holds true. As noticed in [65], their method also applies to hard sphere case. Here is the main result from [65] Theorem 16.6. Assume (293), (294), (296) and (298) for some γ ∈ (−3, 0]. Consider two weak solutios ft , f˜t on some time interval [0, T ]. We assume that they lie in L∞ ([0, T ], P(R3 )) ∩ L1 ([0, T ]; Iγ ). We assume furthermore that for all t ∈ [0, T ], ft (or f˜t ) has a density wrt the Lebesgue measure in R3 . Then there is a constant K = K(κ1 , κ2 , κ3 ) such that for all t ∈ [0, T ] Z t ˜ ˜ W2 (ft , ft ) ≤ W2 (f0 , f0 )exp(K Jγ (fs + f˜s )ds) (299) 0

The following notations are used above. If we denote by P(R3 ) the space of probability measures on R3 , then Z n o P2 (R3 ) ≡ f ∈ P(R3 ), m2 (f ) < +∞ with m2 (f ) ≡ |v|2 f (v)dv. R3

For α ∈ (−3, 0], the space Iα is the space of probability measures such that Z Jα (f ) ≡ sup |v − v∗ |α f (dv∗ ) < +∞. (300) v∈R3

R3

Finally, the Wasserstein distance with quadratic cost on P2 (R3 ) is defined as follows: for g, g˜ in P2 (R3 ), we let H(g; g˜) denote the set of probability measures on R3 × R3 with first marginal g and second marginal g˜. Then R  v ))1/2  W2 (g, g˜) = inf{( R3 ×R3 |v − v˜|G(dv, d˜ (301) R  = min{( R3 ×R3 |v − v˜|G(dv, d˜ v ))1/2 .

By applying this result, uniqueness results follow, see for details [65]. We suggest also reading Section 3 therein, showing a very formal, but always full of ideas, proof of this main result. This paper was further extending previous results by Fournier and Mouhot [67] for moderate angular singularities.

17. Conclusions: Some general remarks and open questions. We end this review by some open questions which certainly are related to our personal feeling. 1) Questions related to the trend to global equilibrium: we refer to paper [55]. Similar studies should now be available in the context of singular Boltzmann equation. 2) Questions related to regularity for similar models, see for instance in Gamba and all [68]. This paper is also interesting for the so called Pozvner’ inequalties, which are important in getting moments estimates. 3) Questions related to comparison principles, see [69]. We have already mentioned a few facts on this topic.

88


4) Questions related to Convolution type inequalities, see the recent study in the cutoff case by Alonso and all. [18]. This is a nice study introducing radial symmetrization and Hardy Young type inequalities, with applications in [19]. 5) Questions related to ideas from Carlen and Gangbo [31], and which are linked with optimal transport theory. 6) Questions related to the use of dispersion estimates [32], which are still not well understood in the context of Boltzmann equation. 7) Questions related to further improvements of 1-D space dimensional problems see [35, 36] in the non cutoff case, and also in the cutoff case [25]. 8) Questions linked to lower bounds, with possible connections with “entropyentropy” methods, see [134] and the references therein. For the spatially inhomogeneous Boltzmann equation, in velocity dimension n, space being the n dimensional torus, Mouhot [109] has obtained quantitative lower bounds estimates, but assuming many assumptions on the solution, which are up to now largely open. In any case, his method applies at least to spatially homogeneous Boltzmann equation in the non cutoff case, and to the inhomogenous Boltzmann equation, in the cutoff case in view of the results of Guo that we have mentioned previously. The assumptions therein are as follows. First of all, B = Φ(|v − v∗ |)b(cos θ)

(302)

∀z ∈ R, cΦ |z|γ ≤ Φ(z) ≤ CΦ |z|γ

(303)

where Φ satisfies one of the following assumptions or the mollified assumption

∀|z| ≥ 1, cΦ |z|γ ≤ Φ(z) ≤ CΦ |z|γ , ||∀|z| ≤ 1, cΦ ≤ Φ(z) ≤ CΦ

(304)

for some constants cΦ , CΦ nonnegative, γ ∈ (−n, 1] and b is a continous positive function on θ ∈ (0, π) such that b(cos θ) sinn−2 θ ∼θ→0 b0 θ−1−ν ,

(305)

for some b0 > 0 and ν ∈ (−∞, 2). For a function f (t, x, v) ≥ 0 on [0, T ] × Tn × Rn , the local density is given by Z ̺f (t, x) ≡ f (t, x, v)dv, Rn

its local energy by

ef (t, x) ≡ its weighted local energy by e′f (t, x) ≡

Z

f (t, x, v)|v|2 dv,

Z

f (t, x, v)|v|γ¯ dv,

Rn

Rn

where γ¯ = (2 + γ)+ , its local entropy by Z hf (t, x) ≡ − f (t, x, v) log f (t, x, v)dv, Rn

p

its local L norm (p ∈ [1, +∞)) by

lfp (t, x) ≡ ||f (t, x, .)||Lp (Rn ) ,

BOLTZMANN EQUATION

89

and its local W 2,∞ norm by wf (t, x) ≡ ||f (t, x, .)||W 2,∞ (Rn ) ,

and finally, the following constants are defined

̺f ≡ inf ̺f (t, x), Ef ≡ sup{ef (t, x) + ̺f (t, x)}, t,x

t,x

Ef′

≡ sup e′f (t, x), t,x Lpf ≡ sup lfp (t, x), t,x

Hf ≡ sup |hf (t, x)|, t,x

Wf ≡ sup wf (t, x). t,x

Since the results of Mouhot also apply to the cutoff case, other assumptions needed are as follows: ̺f > 0, Ef < +∞, Hf < +∞ when γ ≥ 0 and ν < 0. p

Lfγ < +∞, if γ ∈ (−n, 0), where pγ > n/(n + γ).

(306) (307)

(308) Wf < +∞, Ef′ < +∞, when ν ∈ [0, 2) (singular kernels) . Then, the first result from Mouhot gives a lower bound for cutoff kernel, while the second one, of interest for us, deals with the non cutoff case: Theorem 17.1. Assume (302), such that (303) or (304) holds true, with b such that (305), with ν ∈ [0, 2). Let f be a mild solution (using only the transport flow) of the spatially inhomogeneous Boltzmann equation on some time interval (0, T ), T ∈ (0, +∞] such that (i) if Φ satisfies (303) with γ ≥ 0 or if Φ satisfies (304), then f satisfies (306) and (308). (ii) if Φ satisfies (303) with γ < 0, then f satisfies (306), (307) and (308). Then for τ ∈ (0, T ), for any exponent K such that K >2

log (2 +

2ν 2−ν )

log 2

there exists C1 > 0 and C2 > 0 such that ∀t ∈ [τ, T ], ∀x ∈ Tn , ∀v ∈ Rn , f (t, x, v) ≥ C1 e−C2 |v|

K

and in the case ν = 0, K = 2 is allowed.

This result applies at least (for the moment) to the spatially homogeneous solutions. One of the main tool is the use of iteration of the gain operator, together with the decomposition of the collision operator as in Section 3 (rediscovered by Mouhot). Let us sketch his arguments. We start from

Q(g, f ) =

R

Rn

dv∗

R

S n−1

dσBg∗′ (f ′ − f ) + ≡ Q1 + Q2

R

Rn

dv∗

R

S n−1

dσBg∗′ − g∗ )f

(309)

Q2 effect is of course dealt with the Cancellation Lemma, see Section 4 Q2 (g, f ) = S[g]f

where

Z S[g](v) = |S n−2 |(

π/2 0

sinn−2 θ[

1 cosn+γ (θ/2)

− 1]b(θ)dθ)Φ ∗ g

(310)

90


For the analysis of Q1 , it is proven in [109] the following Lemma Lemma 17.2. Assume (302), with Φ sastisfying (303) or (304), and b such that (305), with ν ∈ [0, 2). Then for any smooth fucntions f , g on Rn , one has 1) If Φ satisfies (303) with 2 + γ ≥ 0 or if Φ satisfies (304), then ∀v ∈ Rn , |Q1 (g, f )(v)| . ||g||L1γ¯ ||f ||W 2,∞ < v >γ¯ .

2) If Φ satisfies (303) with 2 + γ < 0, then ∀v ∈ Rn , |Q1 (g, f )(v)| . [||g||L1γ¯ + ||g||Lp ]||f ||W 2,∞ < v >γ¯ , with p > n/(n + γ + 2). In view of Section 7, it is quite clear that this result is not optimal, and certainly there is room for improvement. That also explains why the W 2,∞ was needed in [109]. Anyway, let us give the main steps of the proof. Firstly, by Carleman representation (after echanging the roles of v ′ and v∗′ ), one has Z Z b(cos θ)Φ(v − v∗ ) ′ 1 ′ ′ Q (g, f ) = dv∗ g∗ dv ′ (f − f ), |v − v ′ |n−1 n R Ev,v′ ∗

where Ev,v∗′ denotes the hyperplan orthognal to v − v∗′ and containing v. We use spherical coordinates for v ′ , v ′ = v + ̺σ, where ̺ ∈ R+ and σ ∈ Svn−2 ′ −v ′ , the (n − 2) ∗ unit sphere sphere of Ev,v∗′ , to get Q1 (g, f ) =

Z

Rn

dv∗′ g∗′

Z

+∞

d̺

0

b(cos θ)Φ(v − v∗ ) ̺

Z

Svn−2 ′ −v′

dσ(f (v + ̺σ) − f (v)).

∗

By expanding the difference term with Taylor formula at order two, together with the fact there the first order derivative term vanishes when integrated over the sphere, then one can show that Z |I| ≡ | dσ(f (v + ̺σ) − f (v))| . ̺2 ||f ||W 2,∞ . Svn−2 ′ −v′

∗

Then the rest of the proof follows by duality against test functions in L1 , getting back to the usual variables, and using the changes of variables introduced for the Cancellation lemma. Note the applications of this result to the spatially homogeneous Boltzmann equation without cutoff (Theorem 5.2 of [109]). In fact, let us mention that it also applies (at that time) to solutions constructed by Ukai in the non cutoff case. 10) Questions related to Gevrey regularity. This is a very recent research area in the framework of singular kernels. Apart from the result of Ukai [122] in the spatially inhomogeneous case, let us note the result of propagation of Gevrey regularity by Desvillettes and coll [49]. There is now some recent works in the non cutoff case, see [105, 107]. Furthermore, ChaoJiang Xu has informed us of recent works related to ultra-analycity effects for both Landau and Boltzmann spatially homogeneous equations. 11) Questions related to energy type estimations and applications to global existence issues, and also boundary value problems. We have already mentioned this topic and in particular the series of works in the cutoff case, and also for Landau case, by Guo on one side and by Liu and coll. on the other side. For more on these

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Received September 2009; revised September 2009. E-mail address: [email protected]