LOW MACH NUMBER LIMIT FOR THE ISENTROPIC EULER SYSTEM WITH AXISYMMETRIC INITIAL DATA
arXiv:1109.5339v2 [math.AP] 28 Sep 2011
TAOUFIK HMIDI
Abstract. This paper is devoted to the study of the low Mach number limit for the isentropic Euler system with axisymmetric initial data without swirl. In the first part of the paper we analyze the problem corresponding to the subcritical regularities, that is H s with s > 52 . Taking advantage of the Strichartz estimates and using the special structure of the vorticity we show that the lifespan Tε of the solutions is bounded below by log log log 1ε , where ε denotes the Mach number. Moreover, we prove that the incompressible parts converge to the solution of the incompressible Euler system, when the parameter ε goes to zero. In the second part of the paper we address the same problem 5 2 but for the Besov critical regularity B2,1 . This case turns out to be more subtle at least due to two facts. The first one is related to the Beale-Kato-Majda criterion which is not known to be valid for rough regularities. The second one concerns the critical aspect of the Strichartz estimate L1T L∞ for the acoustic parts (∇∆−1 div vε , cε ): it scales in the space variables like the space of the initial data.
Contents 1. Introduction 2. The functional tool box 2.1. Littlewood-Paley Theory 2.2. Usual and heterogeneous Besov spaces 2.3. Lorentz spaces 3. Energy estimates 4. Strichartz estimates 5. Axisymmetric flows 5.1. Persistence of the geometry 5.2. Dynamics of the vorticity 5.3. Geometric properties 5.4. Some a priori estimates 6. Sub-critical regularities 6.1. Lower bound of Tε 6.2. Incompressible limit 7. Critical regularities 7.1. Proof of Theorem 2 7.2. Logarithmic estimate 7.3. Lower bound of Tε 8. Appendix References
2 6 6 7 10 10 12 14 14 16 17 20 22 22 25 26 28 28 37 40 45
Date: September 30, 2011. 1
1. Introduction The object of this paper is to study the incompressible limit problem for classical solutions of the compressible isentropic Euler equations. The fluid is assumed to evolve in the whole space R3 and it possesses some special geometric property: the flow is invariant under the group of rotations around the vertical axis (Oz). The state of the fluid is described by the velocity field vε and the sound speed cε , through a penalized quasilinear hyperbolic system,
(1.1)
1 ∂t vε + vε · ∇vε + γ¯ cε ∇cε + ε ∇cε = 0 ∂ c + vε · ∇cε + γ¯ cε div vε + 1ε div vε = 0 t ε (vε , cε )|t=0 = (v0,ε , c0,ε ),
with γ¯ a strictly positive number and ε a small parameter called the Mach number. We point out that the derivation of this model can be done from the compressible isentropic equations after rescaling the time and changes of variables, see for instance [14, 18, 24]. This model has been widely considered through the last decades and a special attention is focused on the construction of a family of solutions with a non degenerate time existence. Nevertheless, the most relevant problem is to study rigorously the convergence towards the incompressible Euler equations when the Mach number goes to zero. We recall that the incompressible Euler system is given by, ∂t v + v · ∇v + ∇p = 0 div v = 0 (1.2) v|t=0 = v0 .
The answer to these problems depends on several factors: the domain where the fluid is assumed to evolve: the full space Rd , the torus Td , bounded or unbounded domains. The second factor is the state of the initial data: whether they are well-prepared or not. In the well-prepared case [18, 19], we assume that the initial data are slightly compressible, which means that div v0,ε = O(ε) and ∇c0,ε = O(ε) as ǫ → 0. However, in the ill-prepared case [30], we only assume that the family (v0,ε , c0,ε )ε is bounded in some Sobolev spaces H s with s > d2 + 1 and that the incompressible parts of (v0,ε )ε tend to some field v0 . Remark that in the well-prepared case we have a uniform bound of (∂t vε )ε , and this allows to pass to the limit by using Aubin-Lions compactness lemma. Nevertheless, the ill-prepared case is more subtle because the time derivative ∂t vε , is of size O( 1ε ). To overcome this difficulty, Ukai [30] used the dispersive effects generated by the acoustic waves in order to prove that the compressible part of the velocity and the acoustic term vanish when ε goes to zero. In [14], we deal with a more degenerate case, we allow the initial data to be so ill-prepared that corresponding solutions can tend to a vortex patch or even to a Yudovich solution. Since these solutions do not belong to the Sobolev space H s for any s > 2, we allow the initial data that are not uniformly bounded in these spaces. We point out that this problem has already been studied in numerous papers, see for instance [4, 11, 12, 13, 18, 19, 21, 22, 23, 24, 30]. It is well-known that contrary to the incompressible Euler system, the equations (1.1) develop in space dimension two singularities in finite time and for some smooth initial data, see [25]. This phenomenon holds true for higher dimension, see [27]. On the other hand it was shown in [19] that for the whole space or for the torus domain when the limit system (1.2) exists for some time interval [0, T0 ] then for ever T < T0 there exists ε0 > such that for ε < ε0 the solution to (1.1) lives till the time T . This result was established for the well-prepared case and with sufficient smooth initial data, that is, in H s and s > d2 + 2. We point out that this approach can not be achieved for d2 + 2 > s > d2 + 1. The proof is based on the perturbation theory by estimating in a suitable way the difference between the two systems. As an application in space dimension two, if we denote by Tε the time lifespan of the solution (vε , cε ) then limε→0 Tε = +∞. We observe that we 2
can give an explicit lower bound for the lifespan Tε ≥ C log log 1ε . In [14], we extend these results in the case of Yudovich solutions. In this paper, we try to accomplish the same program in dimension three for axisymmetric initial data. This is motivated by the works of [26, 29] where it is proven that the incompressible Euler system is globally well-posed when the initial data is axisymmetric and belongs to H s , s > 25 . For the definition of the axisymmetry, see Definition 2. The proof relies on the special structure of the vorticity Ω which leads to a global bound of kΩ(t)kL∞ and then we use the Beale-Kato-Majda criterion [6]. Recently, we established in [1] the global well-posedness for (1.2) with initial data 3
+1
p lying in Borderline Besov spaces v0 ∈ Bp,1 , 1 ≤ p ≤ ∞. It is important to mention that in this context the Beale-Kato-Majda criterion is not known to be valid and the geometry is crucially used in different steps of the proof and it is combined with a dynamical interpolation method. Our main goal here is to study the incompressible limit problem for both subcritical and critical cases with ill-prepared axisymmetric initial data. Concerning the subcritical regularities we obtain the following result.
Theorem 1. Let s > that is
5 2
and {(v0,ε , c0,ε )0 52 . We emphasize that in [19] the Sobolev regularity must be larger than 27 . The proof of the Theorem 1 relies on the use of Strichartz estimates for the compressible parts (Qvε ) and the acoustic ones (cε ), see Corollary 2. Thus interpolating this result with the energy estimates, see Proposition 1, we obtain the following result described in Proposition 2: there exists σ > 0 such that (1.3)
k div vε kL1 L∞ + k∇cε kL1 L∞ ≤ C0 εσ (1 + T 2 )eCVε (T ) , T
T
Vε (T ) = k(∇vε , ∇cε )kL1 L∞ . T
It is worthy pointing out that working with the subcritical regularities is very precious to get the preceding inequality. However, this argument fails in the critical spaces as we shall see next. The second ingredient of the proof is the use of the special structure of the vorticity in the axisymmetric case combined with Beale-Kato-Majda criterion. In what follows, we will briefly discuss the main feature of the axisymmetric flows and for the complete computations, see the next sections. By 3
the definition, the velocity takes the form v(x) = v r (r, z)er + v z (r, z)ez in the cylindrical basis and consequently the vorticity Ω := curl v is given by Ω = (∂z v r − ∂r v z )eθ := Ωθ eθ . Therefore, the vorticity dynamics is described by (1.4)
∂t Ωε + vε · ∇Ωε + Ωε div vε =
It follows that the quantity
Ωε r
vεr Ωε . r
satisfies the transport equation:
Ωε = 0. r We observe that this is analogous to the vorticity structure for the compressible Euler equations in space dimensions two. Performing energy estimates we obtain an almost conservation laws:
Ω (1− 1 )k div v k
Ω ε L1 L∞
0,ε
ε t , ∀ p ∈ [1, ∞].
pe p
(t) p ≤ r r L L When the fluid is incompressible we obtain exact conservation laws which are sufficient to lead to the global well-posedness. Concerning the proof of the incompressible limit in the case of ill-prepared initial data, it is done in a straightforward manner by using the Strichartz estimates. In the second part of this paper we will focus on the low Mach number limit for initial data with critical regularities. In our context a function space X is called critical if it is embedded in ∂t + vε · ∇ + div vε
5
2 or more generally Lipschitz class and both have the same scaling, for example Besov space B2,1 3
+1
p , with p ∈ [1, ∞]. These spaces are till now the largest spaces for the local well-posedness to Bp,1 the incompressible Euler equations or more generally for the quasilinear hyperbolic systems of order one. In [1], it is proven that the system (1.2) is globally well-posed for axisymmetric initial data with critical Besov regularity. Therefore it is legitimate to try to accomplish the same program for the system (1.1) as in the subcritical case and especially to quantify a lower bound for the lifespan of the solutions. Nevertheless, to get unfiorm bounds with respect to the parameter ε and remove the penalization term we need to work with critical spaces which are constructed over the Hilbert 5
2 or other modified spaces as we will see next. Although one can prove the local space L2 like B2,1 well-posedness for the system (1.1) with a uniform time existence, the extension of the results of Theorem 1 to the critical case seems to be much more relevant. We distinguish at least two principal difficulties. The first one has a connection with Strichartz estimates (1.3): in the critical framework the quantities kdiv vε kL∞ and kvε k 25 have the same scaling and the interpolation argument used
B2,1
in the proof of Theorem 1 can not work without doing refined improvement. Indeed, we have no P 5 sufficient information about the decay of the remainder series q≥N 2 2 q k(∆q vε , ∆q cε )(t)kL2 and there is no explicit dependence of this decay with respect to the parameters, N, t and ε : it seems that in general the number N can implicitly depend on the variable time and on the parameter ε and this makes the task so hard. To overcome this difficulty, we start with the important observation that 5
5
,Ψ
2 2 any function f ∈ B2,1 belongs to some heterogeneous Besov spaces B2,1 , where Ψ : [−1, +∞[→ R∗+ is a nondecreasing function depending on the profile of f and satisfying limq→+∞ Ψ(q) = +∞. This latter space is defined by the norm X 5 Ψ(q)2 2 q k∆q ukL2 . kuk 52 ,Ψ =
B2,1
q≥−1
Further details and more discussions about these spaces will be found in the next section. We point out that this function Ψ measures the decay of the remainder series in the space of the initial data and we will see that the same decay will occur for the solution uniformly with respect to ε. Therefore one can set up the interpolation argument in the critical framework in the similar way to 4
the subcritical case but without any explicit dependence of the lifespan with respect to the initial data. Concerning the second difficulty, it is related to the Beale-Kato-Majda criterion which is not applicable in the context of critical regularities. In this case the estimate of kΩε (t)kL∞ is not sufficient to propagate the initial regularities and though one should estimate kΩε (t)kB∞,1 instead. We will 0 see next that many geometric properties of the axisymmetric flows are used and play a central role in the critical framework. Our result reads as follows. Theorem 2. Let {(v0,ε , c0,ε )0 0 such that Ψ(q + 1) ≤ C. Ψ(q) q∈N∪{−1} sup
(ii) We define the class U∞ by the set of function Ψ ∈ U satisfying lim Ψ(x) = +∞. x→+∞
s,Ψ (iii) Let s ∈ R, p, r ∈ [1, +∞] and Ψ ∈ U . We define the heterogeneous Besov space Bp,r as follows:
s,Ψ u ∈ Bp,r ⇐⇒ kukB s,Ψ = Ψ(q)2qs k∆q ukLp
ℓr
p,r
< +∞.
s,Ψ Remarks 2. (1) We observe that when we take Ψ(x) = 2αx with α ∈ R+ , the space Bp,r reduces s+α to the classical Besov space Bp,r . s,Ψ (2) The condition (2) seems to be necessary for the definition of Bp,r : it allows to get a definition which is independent of the choice of the dyadic partition.
The following lemma is important for the proof of Theorem 2. Roughly speaking, we will prove that any element of a given Besov space is always more regular than the prescribed regularity. s . Then there exists a function Ψ Lemma 2. Let s ∈ R, p ∈ [1, +∞], r ∈ [1, +∞[ and f ∈ Bp,r s,Ψ belonging to U∞ such that f ∈ Bp,r .
Proof. We observe that the proof P reduces to the following statement: assume that a strictly positive sequence (cq )q≥−1 satisfies q cq < +∞, then there exists a nondecreasing sequence (aq )q≥−1 satisfying (2.2)
lim aq = +∞,
q→+∞
aq+1 ≤C q∈N∪{−1} aq sup
and such that X
(2.3)
aq cq < +∞.
q≥−1
Let bq =
P
n≥q cn
− 1 2
, then (bq )q≥−1 is a nondecreasing sequence going to infinity. Moreover X
(2.4)
q≥−1
bq cq ≤ 2
X
q≥−1
cq
1 2
.
Indeed, we introduce the piecewise function f : [−1, +∞[→ R+ defined by f (x) = cq ,
for
x ∈ [q, q + 1[ and 8
q ∈ N ∪ {−1}.
Then we get by obvious computations, X
bq cq =
q≥−1
X
q≥−1
≤
X Z
R q+1 q
R +∞ q q+1
q≥−1 q Z +∞
f (x)dx 1 f (x)dx 2
f (x) R +∞ 1 dx 2 f (y)dy x
f (x) R +∞ 1 dx 2 −1 f (y)dy x Z +∞ 1 X 1 2 ≤ 2 f (x)dx = 2 cq 2 . ≤
−1
q≥−1
Now we will construct by recursive procedure a sequence (aq )q≥−1 satisfying (2.2) and (2.3), and such that X 1 X cq 2 . aq cq ≤ 2 q≥−1
q≥−1
Let (aq )q≥−1 be the sequence defined by the following recursive formula aq+1 = 12 aq + min(bq+1 , 2aq ) (2.5) a−1 = b−1 .
We will show first that aq ≤ bq . This is true for q = −1 and since (bq )q≥−1 is nondecreasing then
1 1 (aq − bq+1 ) ≤ (aq − bq ). 2 2 Thus we find by the principle of recurrence that aq ≤ bq , ∀q ≥ −1. From this property and (2.4) we P get the convergence of the series q≥−1 aq cq and more precisely X 1 X cq 2 . aq cq ≤ 2 aq+1 − bq+1 ≤
q≥−1
q≥−1
Let us now prove that the sequence (aq )q≥−1 is nondecreasing. Indeed, by easy computations and from the nondecreasing property of (bq )q≥−1 and the fact aq ≤ bq we get 1 min(bq+1 , 2aq ) − aq aq+1 − aq = 2 1 min(bq+1 − aq , aq ) = 2 1 ≥ min(bq − aq , aq ) ≥ 0. 2 Now we will prove that (aq )q≥−1 converges to +∞, otherwise it will converge to a finite real number ℓ > 0. Using the relation (2.5) combined with the fact that limq→+∞ bq = +∞ yields necessary to ℓ = 32 ℓ, which contradicts the fact that ℓ ∈]0, +∞[. On the other hand, we have 1≤
aq+1 3 ≤ · aq 2
This ends the proof of all the properties of the sequence (aq )q≥−1 .
Remark 2. FromP the proof of Lemma 2 we may easily check that we can replace the value of bq by any expression ( n≥q cn )−α , with α < 1. The case α = 1 is not true, at least for any convergent geometric series. 9
As a consequence we get the following result. Corollary 1. Let s ∈ R, p ∈ [1, ∞], r ∈ [1, ∞[ and (fε )0 0, α ≥ 0, r ∈ [1, +∞], there exists C > 0 such that k(vε , cε )(t)kB s,Ψ ≤ Ck(v0,ε , c0,ε )kB s,Ψ eCVε (t) 2,r
2,r
with Vε (t) := k∇vε kL1t L∞ + k∇cε kL1t L∞ .
s . The above estimate holds true for homogeneous Besov spaces B˙ 2,r
Proof. (1) Taking the L2 inner product of the first equation of (1.1) with vε and integrating by parts Z Z 1d 1 1 2 2 2 div vε (|vε | + γ¯cε )dx − cε div vε dx = 0. kvε (t)kL2 − 2 dt 2 R3 ε R3 Multiplying the second equation of (1.1) by cε and integrating by parts Z Z 1d 1 1 2 2 (div vε )cε dx + kcε (t)kL2 + (¯ γ− ) cε div vε dx = 0. 2 dt 2 R3 ε R3 Thus summing up these identities yields to Z d div vε |vε |2 + (1 − γ¯)c2ε dx kvε (t)k2L2 + kvε (t)k2L2 = dt R3 . k div vε (t)kL∞ kvε (t)k2L2 + kcε (t)k2L2 .
The result follows easily from Gronwall’s lemma. (2) We will use localize in frequency the equations and use some results on the commutators. Let q ≥ −1 and set fq := ∆q f , then ( 1 ∂t vε,q + vε · ∇vε,q + γ¯ cε ∇cε,q + 1ε ∇cε,q = −[∆q , vε · ∇]vε − γ¯ [∆q , cε ]∇cε := Tε,q (3.1) 2 ∂t cε,q + vε · ∇cε,q + γ¯ cε div vε,q + 1ε div vε,q = −[∆q , vε · ∇]cε − γ¯ [∆q , cε ] div vε := Tε,q Denote wε,q (t) := |cε,q (t)|2 + |vε,q (t)|2 , then taking the L2 inner product in a similar way to the first part (1) we get Z Z 1 d 1 cε ∇cε,q · vε,q + cε,q div vε,q dx div vε wε,q dx − γ¯ kwε,q (t)kL1 = 2 dt 2 3 R3 Z R 1 2 (Tε,q vε,q + Tε,q cε,q )dx + 3 R Z Z Z 1 1 2 = (Tε,q vε,q + Tε,q cε,q )dx ∇cε · cε,q vε,q )dx + div vε wε,q dx + γ¯ 2 R3 3 3 R R 1 1 2 ∞ ∞ , Tε,q )kL2 kwε,q (t)kL2 1 . . k div vε (t)kL + k∇cε (t)kL kwε,q (t)kL1 + k(Tε,q We have used in the last line the Young inequality |ab| ≤ 12 (a2 + b2 ). It follows that d 1 2 , Tε,q )kL2 . k(vε,q , cε,q )(t)kL2 . k div vε (t)kL∞ + k∇cε (t)kL∞ k(vε,q , cε,q )(t)kL2 + k(Tε,q dt Multiplying by 2qs Ψ(q) and summing up over q we get d 1 2 k(vε , cε )(t)kB s,Ψ ≤ k div vε (t)kL∞ + k∇cε (t)kL∞ k(vε , cε )(t)kB s,Ψ + 2qs Ψ(q)k(Tε,q , Tε,q )kL2 r . 2,r 2,r dt ℓ Now according to the Lemma 6 we have: for s > 0, r ∈ [1, ∞] and Ψ ∈ U 2qs Ψ(q)k[∆q , v · ∇]ukL2 ℓr . k∇vkL∞ kukB s,Ψ + k∇ukL∞ kvkB s,Ψ . 2,r
11
2,r
This yields to d k(vε , cε )(t)kB s,Ψ . k∇vε (t)kL∞ + k∇cε (t)kL∞ k(vε , cε )(t)kB s,Ψ . 2,r 2,r dt It suffices to use Gonwall’s inequality to get the desired estimate.
4. Strichartz estimates The main goal of this section is to establish some Strichartz estimates for the compressible and acoustic parts which are governed by coupling nonlinear wave equations. As it was shown in the pioneering work of Ukai [30] the use of the dispersion is crucial to get rid of the well-prepared assumption when studying the convergence towards the incompressible Euler system. More precisely, it was shown that by averaging in time the compressible and acoustic parts vanish when the Mach number goes to zero. In our case we will take advantage of the Strichartz estimates firstly to deal with the ill-prepared case and secondly to improve the lifespan by a judicious combination with the special structure of the vorticity for the axisymmetric flows. The results concerning Strichartz estimates that we will use here are well-known in the literature and are shortly discussed. Nevertheless we will give more details about their applications for the isentropic Euler system. We will start with rewriting the system (1.1) with the aid of the free wave propagator ∂t vε + 1ε ∇cε = fε := −vε · ∇vε − cε ∇cε (4.1) ∂t cε + 1ε div vε = gε := −vε · ∇cε − cε div vε .
−1 Let Q be the operator √ Qv := ∇∆ div v which is nothing but the compressible part of the velocity v. Set |D| := −∆, then by simple computations we show that the quantity
Γǫ := Qvε − i∇|D|−1 cε
satisfies the wave equation
i ∂t Γǫ + |D|Γǫ = Qfε − i∇|D|−1 gε . ε
(4.2) Analogously, the quantity
γǫ := |D|−1 div vε + i cε
satisfies the wave equation
i ∂t γǫ + |D|γǫ = |D|−1 div fε + i gε . ε We will use the following Strichartz estimates which can be found for instance in [5, 16].
(4.3)
Lemma 3. Let ϕ be a complex solution of the wave equation i ∂t ϕ + |D|ϕ = F. ε Then for every s ∈ R, r > 2 there exists C such that for every T > 0 we have 1 kϕk r s− 23 + 1r ≤ Cε r kϕ(0)kB˙ s + kF kL1 B˙ s . LT B˙ ∞,1
2,1
T
2,1
As a consequence we get the following Strichartz estimates for both compressible and acoustic parts. 5
−1
2 r Corollary 2. Let r > 2 and v0,ε , c0,ε ∈ B2,1 . Then the solutions of the system (1.1) satisfy for all T > 0 and 0 < ε ≤ 1,
kQvε kLr B˙ 0 T
∞,1
+ kcε kLr B˙ 0 T
1
∞,1
12
≤ C0 ε r (1 + T )eCVε (T )
with C0 a constant depending on r and the norm k(v0,ε , c0,ε )k
5−1 r
2 B2,1
Vε (T ) =
Z
T 0
and
(k∇vε (t)kL∞ + k∇cε (t)kL∞ dt.
Proof. Applyiny Lemma 3 to the equation (4.2) with r > 2 and s = kΓε kLr B˙ 0
∞,1
T
≤ Cε
1 r
kΓε (0)k
2−r B˙ 2,1
−
1 r
we get + kQfε − i∇|D|−1 gε k 1 32 − r1 .
1
3
3 2
LT B˙ 2,1
Since the operators Q and ∇|D|−1 are homogeneous of order zero then they are continuous on the homogeneous Besov spaces and thus we get 1 ≤ Cε r k(v0,ε , c0,ε )k 23 − 1r + k(fε , gε )k 1 23 − 1r kΓε kLr B˙ 0 LT B˙ 2,1
B˙ 2,1
∞,1
T
≤ Cε
1 r
k(v0,ε , c0,ε )k
3 1 2−r B2,1
+ k(fε , gε )k
3 1 2−r L1T B2,1
.
s ֒→ B ˙ s , for s > 0. To estimate the We point out that we have used above the embedding B2,1 2,1 terms fε and gε we will use the following law product that can be easily proven by using Bony’s decomposition: for s > 0 s kf ∂j gkB2,1 . kf kL∞ kgkB s+1 + kgkL∞ kf kB s+1 . 2,1
2,1
Therefore we find k(fε , gε )k
≤ Ck(vε , cε )kL∞ k(vε , cε )k
3−1 r
2 B2,1
5−1 r
2 B2,1
5
− 1r
2 Combining this estimate with Sobolev embedding B2,1
k(fε , gε )k
3−1 r
2 B2,1
.
֒→ L∞ (true for r ≥ 1), we get
≤ Ck(vε , cε )k2
5−1 r
2 B2,1
.
Applying Proposition 1 yields to k(vε , cε )k
5−1 r
2 L∞ T B2,1
Thus we obtain for r > 2, kΓε kLr B˙ 0 T
∞,1
≤ Ck(v0,ε , c0,ε )k
5−1 r
2 B2,1
1 ≤ Cε r k(v0,ε , c0,ε )k
3 1 2−r B2,1
eCVε (T ) .
+ k(v0,ε , c0,ε )k2
5−1 2 r B2,1
T eCVε (T )
1
≤ C0 ε r (1 + T )eCVε (T ) .
Since the real part of Γε is the compressible part of vε then kQvε kLr B˙ 0
1
∞,1
T
≤ C0 ε r (1 + T )eCVε (T ) .
By the same way we prove a similar result for γε : kγε kLr B˙ 0 T
∞,1
1
≤ C0 ε r (1 + T )eCVε (T ) .
This gives the desired estimate for kcε kLr B˙ 0 . T
∞,1
Proposition 2. Let (v0,ε , c0,ε ) be a H s -bounded family with s > 52 and vε , cε ∈ C([0, Tε ); H s ) be the maximal solution of the system (1.1). Then for every r > 2, 0 ≤ T < Tε and ε ∈]0, 1] we have, 2s−5
≤ C0 ε r(2s−3) (1 + T 2 )eCVε (T ) . + k∇cε kL1 B∞,1 k div vε kL1 B∞,1 0 0 T
T
13
Proof. Applying Lemma 4 with ϕ = Qvε and using the identity div Qvε = div vε , we get ≤ CT
k div vε kL1 B∞,1 0 T
≤ CT
2s−5 1− r(2s−3)
2s−5 1− r(2s−3)
2s−5
2
2s−3 kQvε kL2s−3 r L∞ kQvε kL∞ H s T 2s−5 2s−3 0 LrT B˙ ∞,1
kQvε k
T 2 2s−3 s L∞ T H
kvε k
.
0 ֒→ L∞ . Now, combining this estimate We have used in the last inequality the embedding B˙ ∞,1 with Proposition 1 and Corollary 2,
k div vε kL1 B∞,1 0 T
2s−5
2s−5
1
≤ C0 ε r(2s−3) (1 + T )1+ 2s−3 (1− r ) eCVε (t) 2s−5
≤ C0 ε r(2s−3) (1 + T 2 )eCVε (t) .
. Similarly we obtain an analogous estimate for the acoustic part k∇cε kL1 B∞,1 0 T
5. Axisymmetric flows In this section we intend to establish some preliminary results about the axisymmetric geometry for the compressible flows. First, we prove the persistence in time of this geometry when it is initially prescribed and second we analyze the structure and the dynamics of the vorticity. We end this section with some useful a priori estimates. 5.1. Persistence of the geometry. The study of axisymmetric flows was initiated by Ladyzhanskaya [20] and Ukhoviskii and Yudovich [29] for both incompressible Euler and Navier-Stokes equations. This study has been recently extended for other models of incompressible fluid dynamics like stratified Euler and Navier-Stokes systems [1, 17]. First of all we will show the compatibility of this geometry with the model (1.1) but we need before to give a precise definition of axisymmetric vector fields. Definition 2. • We say that a vector field v : R3 → R3 is axisymmetric if it satisfies R−α {v(Rα x)} = v(x),
∀α ∈ [0, 2π],
∀x ∈ R3 ,
where Rα denotes the rotation of axis (Oz) and with angle θ. An axisymmetric vector field v is called without swirl if its angular component vanishes, which is equivalent to the fact that v takes the form: 1 v(x) = v r (r, z)er + v z (r, z)ez , x = (x1 , x2 , z), r = (x21 + x22 ) 2 , where er , eθ , ez is the cylindrical basis of R3 and the components v r and v z do not depend on the angular variable. • A scalar function f : R3 → R is called axisymmetric if the vector field x 7→ f (x)ez is axisymmetric, which means that f (Rα x) = f (x), ∀ x ∈ R3 , ∀ α ∈ [0, 2π]. This is equivalent to say that f depends only on r and z. Now we will prove the persistence of this geometry for the system (1.1). Proposition 3. Let (v0,ε , c0,ε ) be a smooth axisymmetric initial data without swirl. Then the associated maximal solution of (1.1) remains axisymmetric. Proof. For the sake of the simplicity we will remove the subscript ε from our notations. We set vα (t, x) = R−α {v(t, Rα x)}
and cα (t, x) = c(t, Rα x).
Our goal is to show that (vα , cα ) solves the system (1.1). First of all, we claim that (5.1)
(vα · ∇vα )(t, x) = R−α {(v · ∇v)(t, Rα x)}. 14
Indeed, obvious computations yield to (vα · ∇vα )(t, x)
=
=
3 X
i R−α {v(t, Rα x)} ∂i {R−α v(t, Rα x)}
i=1 3 X
i R−α {v(t, Rα x)} R−α ∂i {v(t, Rα x)}
i=1
=
R−α
3 X
i R−α {v(t, Rα x)} ∂i {v(t, Rα x)}
i=1
:= R−α w.
From the formula
i ∂i {v(t, Rα x)}j = R−α {(∇v j )(t, Rα x)} and using the fact that the rotations preserve Euclidian scalar product we obtain w
j
=
3 X i=1
i i R−α {v(t, Rα x)} R−α {(∇v j )(Rα x)}
= (v · ∇v j )(t, Rα x).
Therefore we get
(vα · ∇vα )(t, x) = R−α {(v · ∇v)(t, Rα x)}.
Now if f is a scalar function then (5.2)
R−α {(∇f )(Rα x)} = ∇{f (Rα x)}.
Using this identity and (5.1) we prove that (vα , cα ) satisfies the first equation of (1.1). It remains to prove that this couple of functions satisfies also the second equation of (1.1). We write according to the identity (5.2) and from the fact that Rα is an isometry, (vα · ∇cα )(t, x) = It remains to check that Indeed, set R−α
3 X i=1
R−α {v(t, Rα x)} · R−α {∇c}(t, Rα x)
= {v · ∇c}(t, Rα x).
div vα (t, x) = {div v}(t, Rα x). := (aij )1≤i,j≤3 , then since R⋆α = R−α , we find div vα (t, x) =
3 X
i,j=1
=
∂i {aij v j (t, Rα x)}
3 X
i,k,j=1
=
3 X
k,j=1
aij aik {∂k v j }(t, Rα x)
δkj {∂k v j }(t, Rα x)
= {div v}(t, Rα x).
Finally, we deduce that (vα , cα ) satisfies the same equations of (v, c) and by the uniqueness of the solutions we get vα = v, cα = c for every α ∈ [0, 2π] and thus the solution is axisymmetric. To 15
achieve the proof it remains to show that the angular component v θ of the velocity v is zero. By direct computations using the axisymmetry of the solution (v, c) we get vr ∂t v θ + v · ∇v θ + v θ = 0. r It follows from the maximum principle and Gronwall’s inequality that kvr /rkL1 L∞
kv θ (t)kL∞ ≤ kv0θ kL∞ e
t
.
Therefore if v0θ ≡ 0 then v θ (t) ≡ 0 everywhere the solution is defined.
Remark 3. From the above computations we get the following assertions: (1) If c is a scalar axisymmetric function then its gradient ∇c is an axisymmetric vector field. (2) If v is an axisymmetric vector field, then its divergence div v is an axisymmetric scalar function. 5.2. Dynamics of the vorticity. We will start with recalling some algebraic properties of the axisymmetric vector fields and especially we will discuss the special structure of the vorticity of the system (1.1). First, we make some general statements: let w = wr (r, z)er + wθ (r, z)eθ + wz (r, z)ez and v = v r (r, z)er + v z (r, z)ez be two smooth vector fields, then vr wθ ∂θ + wz ∂z , div v = ∂r v r + + ∂z v z . r r Easy computations show that the vorticity Ω := curl v of the vector field v takes the form,
(5.3)
w · ∇ = wr ∂r +
Ω = (∂z v r − ∂r v z )eθ .
Now let us study under this special geometry the dynamics of the vorticity of the system (1.1). Denote by Ωε = (∂z vεr − ∂r vεz )eθ := Ωθε eθ , the vorticity of vε . Then applying the curl operator to the velocity equation yields ∂t Ωε + curl(vε · ∇vε ) = 0. By straightforward computations we get the identity Therefore we get
curl(vε · ∇vε ) = vε · ∇Ωε − Ωε · ∇vε + Ωε div vε .
∂t Ωε + vε · ∇Ωε + Ωε div vε = Ωε · ∇vε Now, since Ωε = Ωθε eθ and by (5.3) it follows that the stretching term takes the form, 1 Ωε · ∇vε = Ωθε ∂θ (vǫr er + vǫz ez ) r vr = Ωθε ǫ eθ r vǫr = Ωε . r Consequently the vorticity equation becomes vr (5.4) ∂t Ωε + vε · ∇Ωε + Ωε div vε = ǫ Ωε . r Ωε Thus the quantity r is governed by the following equation Ωε = 0. (5.5) (∂t + vε · ∇ + div vε ) r We observe that this equation is analogous to the vorticity equation in dimension two, In the case of incompressible axisymmetric flows the quantity Ωrε is just transported by the flow and this gives new conservation laws that lead to global existence of smooth solutions. However in our case we can 16
not get global estimates due to the presence of div vε , yet we have already seen that this quantity is damped by higher oscillations and thus we can expect that the time lifespan can grow when the Mach number becomes small. This will be clearly discussed later in the next sections. 5.3. Geometric properties. When we deal with the critical regularities, it seems that the only use of the equation (5.5) is not sufficient for our study and so we need more refined properties of the vorticity. We start with the following results. Proposition 4. Let v = (v 1 , v 2 , v 3 ) be a smooth axisymmetric vector field without swirl. Then the following assertions hold true. (1) The vector ω = ∇ × v = (ω 1 , ω 2 , ω 3 ) satisfies ω(x) × eθ (x) = (0, 0, 0). In particular, we have for every (x1 , x2 , z) ∈ R3 , ω 3 = 0,
x1 ω 1 (x1 , x2 , z) + x2 ω 2 (x1 , x2 , z) = 0
and
ω 1 (x1 , 0, z) = ω 2 (0, x2 , z) = 0. (2) For every q ≥ −1, ∆q u is axisymmetric without swirl and
(∆q u1 )(0, x2 , z) = (∆q u2 )(x1 , 0, z) = 0.
(3) Let Ω be a free divergence vector field such that Ω(x) × eθ (x) = 0. Then for q ≥ −1 we have ∆q Ω(x) × eθ (x) = 0.
Proof. The results (1) − (2) and (3) are proved for example in [1]. It remains to prove the last assertion. First, it is easy to check that (∆q Ω)3 = ∆q (Ω3 ) and thus the last component of ∆q Ω is zero and consequently our claim reduces to the following identity, x1 ∆q Ω1 + x2 ∆q Ω2 = 0. Using Fourier transform this is equivalent to c2 (ξ) = 0. c1 (ξ) + ∂2 ϕ(2−q |ξ|)Ω ∂1 ϕ(2−q |ξ|)Ω
Straightforward computations and the fact that Ω3 ≡ 0 yield c2 (ξ) c1 (ξ) + ξ 2 Ω c2 (ξ) = 2−q |ξ|−1 ϕ′ (2−q |ξ|) ξ 1 Ω c1 (ξ) + ∂2 ϕ(2−q |ξ|)Ω ∂1 ϕ(2−q |ξ|)Ω c1 (ξ) + ∂2 Ω c2 (ξ) + ϕ(2−q |ξ|) ∂1 Ω [ = −i2−q |ξ|−1 ϕ′ (2−q |ξ|)div Ω(ξ) −q 1 − iϕ(2 |ξ|)F x1 Ω + x2 Ω2 (ξ) = 0.
The last identity is an easy consequence from the hypotheses div Ω = 0 and Ω × eθ = 0.
The next result deals with some properties of the flow ψ associated to a time-dependent axisymmetric flow (t, x) 7→ v(t, x). It is defined as follow through the integral equation Z t v τ, ψ(τ, x) dτ. ψ(t, x) = x + 0
It is well-known from the theory of differential equation that if I is an interval and v ∈ L1 (I, Lip) then the flow is well-defined on the full interval I. Denote by G the group of rotations with axis (Oz), that is G := Rθ , θ ∈ [0, 2π] . Now we will prove the following result. 17
Proposition 5. Let (t, x) 7→ v(t, x) be a time-dependent smooth axisymmetric vector field without swirl and (t, x) 7→ ψ(t, x) its flow. Then the following results hold true. (1) For all Rθ ∈ G we have ∀x ∈ R3 .
Rθ (ψ(t, x)) = ψ(t, Rθ x),
(2) For every x ∈ R3 , t ≥ 0 we have
ψ(t, x) · eθ (x) = 0.
(3) For every t, the vector field x 7→ v(t, ψ(t, x)) is axisymmetric without swirl. (4) For all q ∈ N we have Sq v(t) ◦ ψ(t) (x) · eθ (x) = 0, ∀x ∈ R3 .
Proof. (1) We set ψθ (t, x) := Rθ ψ(t, x). Thus differentiating with respect to t we get ∂t ψθ (t, x) = Rθ v(t, ψ(t, x)) .
Since v is axisymmetric then and consequently
Rθ v(t, X) = v(t, Rθ X),
∀X ∈ R3
∂t ψθ (t, x) = v t, Rθ (ψ(t, x)) = v(t, ψθ (t, x)),
ψθ (0, x) = Rθ x.
It is easy to see that (t, x) 7→ ψ(t, Rθ x) satisfies the same differential equation of ψθ and thus by uniqueness we get the desired identity. (2) This result, which means that the trajectory of a given particle x remains in the vertical plan, was proved in Proposition [2]. (3) Let Rθ ∈ G. Since v is axisymmetric then using (1) of Proposition 5 yields to Rθ v(t, ψ(t, x)) = v(t, Rθ (ψ(t, x))) = v(t, ψ(t, Rθ x)).
It remains to show that the angular component of the vector field x 7→ v(t, ψ(t, x)) is zero. Since the angular component of v is zero then v(t, X) · eθ (X) = 0,
∀X ∈ R3
and by (2) of Proposition 5 we have eθ (ψ(t, x)) = ±eθ (x). It follows that the angular component of the vector field x 7→ v(t, ψ(t, x)) is vanishes and consequently it is axisymmetric without swirl. (4) Combining the preceding result with the part (2) of Proposition 4 we obtain that the vector field x 7→ Sq v(t) ◦ ψ(t)) (x) is also axisymmetric without swirl and consequently its angular component is zero. The end of this section is devoted to the study of some geometric properties of a compressible transport model. It is a vorticity equation like in which no relations between the vector field v and the solution Ω are supposed. ∂t Ω + (v · ∇)Ω + Ω div v = Ω · ∇v, (CT) Ω|t=0 = Ω0 .
We will assume that v is axisymmetric and the unknown function Ω = (Ω1 , Ω2 , Ω3 ) is a vector field. The following result describes the persistence of some initial geometric conditions of the solution Ω. Proposition 6. Let T > 0 and v be an axisymmetric vector field without swirl and belonging to the space L1 ([0, T ], Lip(R3 )). Denote by Ω the unique solution of (CT) corresponding to smooth initial data Ω0 . Then we have the following properties: (1) If div Ω0 = 0, then div Ω(t) = 0, ∀ t ∈ [0, T ]. 18
(2) If Ω0 × eθ = 0, then we have
∀t ∈ [0, T ],
Ω(t) × eθ = 0.
Consequently, Ω1 (t, x1 , 0, z) = Ω2 (t, 0, x2 , z) = 0, and
vr Ω. r Proof. (1) We apply the divergence operator to the equation (V) ∂t Ω + (v · ∇)Ω + Ω div v =
∂t div Ω + div (u · ∇Ω + Ω div u) = div (Ω · ∇u).
Straightforward computations yield to
3 X
div (u · ∇Ω) = u · ∇div Ω +
∂i uj ∂j Ωi
i,j=1
div (Ω div u) = div Ω div u + Ω · ∇div u 3 X
div (Ω · ∇u) = Ω · ∇div u +
∂i uj ∂j Ωi .
i,j=1
Thus, the quantity div Ω satisfies the equation
∂t div Ω + u · ∇div Ω + div Ω div u = 0.
Using maximum principle and Gronwall inequality we get
kdiv ukL1 L∞
kdiv Ω(t)kL∞ ≤ div Ω0 kL∞ e
t
.
Therefore if div Ω0 = 0 then for every time Ω(t) remains incompressible. (2) We denote by (Ωr , Ωθ , Ωz ) the coordinates of Ω in cylindrical basis. It is obvious that Ωr = Ω·er . Recall that in cylindrical coordinates the operator u · ∇ has the form 1 u · ∇ = ur ∂r + uθ ∂θ + uz ∂z = ur ∂r + uz ∂z . r We have used in the last equality the fact that for axisymmetric flows the angular component is zero. Hence we get (u · ∇Ω) · er = ur ∂r Ω · er + uz ∂z Ω · er = (ur ∂r + uz ∂z )(Ω · er ) = u · ∇Ωr ,
Where we use ∂r er = ∂z er = 0. Now it remains to compute (Ω · ∇u) · er . By a straightforward computations we get, 1 (Ω · ∇u) · er = Ωr ∂r u · er + Ωθ ∂θ u · er + Ω3 ∂3 u · er r r r 3 = Ω ∂r u + Ω ∂3 ur . Thus the component Ωr obeys to the equation ∂t Ωr + u · ∇Ωr + div u Ωr = Ωr ∂r ur + Ω3 ∂3 ur .
From the maximum principle we deduce Z t r kΩr (τ )kL∞ + kΩ3 (τ )kL∞ k∇u(τ )kL∞ dτ. kΩ (t)kL∞ ≤ 0
On the other hand the component Ω3 satisfies the equation
∂t Ω3 + u · ∇Ω3 + div u Ω3 = Ω3 ∂3 u3 + Ωr ∂r u3 . 19
This leads to 3
kΩ (t)k
L∞
≤
Z
0
t
kΩ3 (τ )kL∞ + kΩr (τ )kL∞ k∇u(τ )kL∞ dτ.
Combining these estimates and using Gronwall’s inequality we obtain for every t ∈ R+ , Ω3 (t) = Ωr (t) = 0, which is the desired result. Under these assumptions the stretching term becomes 1 θ Ω · ∇u = Ω ∂θ (ur er ) r 1 1 r θ u Ω eθ = ur Ω. = r r 5.4. Some a priori estimates. Our goal in this section is to establish some a priori estimates that will be used later for both cases of critical and sub-critical regularities. Our result reads as follows. Proposition 7. Let (vε , cε ) be a smooth axisymmetric solution of the system (1.1). Then the following estimates hold true. (1) Denote by Ωε the vorticity of vε , then for t ≥ 0 and p ∈ [1, ∞] we have
Ω
Ω0
ε
(1− 1 )kdiv vε kL1t L∞
(t) p ≤ ε p e p r r L L and for 1 < p < ∞, q ∈ [1, +∞]
Ω0
Ω (1− 1 )kdiv vε kL1 L∞
ε
ε t . (t) ≤ C
p,q
p,q e p
r r L L (2) For t ≥ 0, we have
Ω0 vr k div vε kL1 L∞
t k ε kL∞ . kvε kL2 + k div vε kB∞,1 + ε 3,1 e . 0 r r L (3) For t ≥ 0, we have o n k div v k Ckvε kL1 L2 C k div vε kL1 B0
kΩε (t)kL∞ ≤ kΩ0ε kL∞ e
t
e
t
∞,1
+t kΩ0ε /rkL3,1 e
ε
∞ L1 tL
.
Proof. (1) To prove this result we will use the characteristic method. Let ψε denote the flow associated to the velocity vε and defined by the integral equation Z t vε (τ, ψε (τ, x))dτ. ψε (t, x) = x + 0
Set fε (t, x) =
Ωε r (t, ψ(t, x)),
then it is easy to see from the equation (5.5) that ∂t fε (t, x) + (div vε )(t, ψ(t, x))fε (t, x) = 0.
Thus we get easily (5.6)
fε (t, x) = fε (0, x) e−
Rt 0
(div vε )(τ,ψ(τ,x))dτ
For p = +∞ we get
Ω0
Ω
ε
(t) ∞ = ε ∞ . r r L L Now for p ∈ [1, +∞[ we will use the identity Rt
det∇ψε (t, x) = e
0 (div
20
vε )(τ,ψ(τ,x))dτ
.
and thus we get by (5.6)
Ω
ε p
(t) p r L
= kfε (t) ◦ ψε−1 (t, ·)kpLp Z |fε (t, x)|p det∇ψε (t, x)dx = R3 Z Rt |fε (0, x)|p e(1−p) 0 (div vε )(τ,ψ(τ,x)) dτ = R3
Ω0 p (p−1)kdiv v k ε L1 L∞
t ≤ ε pe . r L The estimates in Lorentz spaces are a direct consequence of real interpolation argument. r (2) The estimate of the quantity k vrǫ kL1t L∞ will require the use of some special properties of axisymmetric flows. First, we split the velocity into compressible and incompressible parts: vε = Pvε + ∇∆−1 div vε . We point out that in this decomposition both vector fields are also axisymmetric. Indeed, from Remark 3, the scalar function div vε is an axisymmetric scalar function and ∆−1 div vε too. Again from Remark 3 the vector field ∇∆−1 div vε is axisymmetric. Now obvious computations yield to vεr = (Pvε )r + (∇∆−1 div vε ) · er = (Pvε )r + ∂r ∆−1 div vε .
(5.7) Therefore we deduce
vεr (Pvε )r ∂r = + ∆−1 div vε · r r r To estimate the second term of the preceding identity we make use of the algebraic identity described in Lemma 7 and dealing with the action of the operator ∂rr ∆−1 u over axisymmetric functions.
∂
r −1 ∆ div v
ε ∞ r L
≤ ≤
2 X
i,j=1 2 X
i,j=1
kRij div vε kL∞ kRij div vε kB˙ 0
∞,1
≤ Ck div vε 0 ˙
B∞,1
≤ Ckvε kL2
+ k div vε
0 B∞,1
.
0 ֒→ L∞ combined with the continuity of Riesz transforms on We have used the emebedding B˙ ∞,1 the homogeneous Besov spaces. Furthermore, In the last line we use Bernstein inequalities in order to bound low frequencies: X X 5 ˙ j div vε kL∞ ≤ Ckvε kL2 k∆ 2 2 j ≤ Ckvε kL2 . j∈Z−
j∈Z−
Now let us come back to (5.7) and look at the incompressible term. Since Pvε is axisymmetric and satisfying furthermore div Pvε = 0 and curl Pvε = Ωε then we can use the following inequality proven by Shirota and Yanagisawa [26], θ Z (Pv )r | Ωrε (y)| ε (x) ≤ C dy. 2 r R3 |x − y| 21
Now since
1 |·|2
belongs to Lorentz space L3,∞ then the usual convolution laws yield
(Pv )r
Ω
ε ε (t) ∞ . (t) 3,1 .
r r L L
According to the first part of Proposition 7 we have
Ω0
Ω k div vε kL1 L∞
ε t .
(t) 3,1 ≤ C ε 3,1 e r r L L
Combining these estimates we get (5.8)
k
Ω0 vεr k div vε kL1 L∞
t + ε 3,1 e kL∞ . kvε kL2 + k div vε kB∞,1 . 0 r r L
(3) Applying maximum principle to the vorticity equation (5.4) and using Gronwall’s inequality we get k div vε kL1 L∞ +k
(5.9)
kΩε (t)kL∞ ≤ kΩε (0)kL∞ e
t
vǫr r
kL1 L∞ t
.
Plugging the estimate (5.8) into (5.9) gives n
Ckvε kL1 L2 C k div vε kL1 B0
kΩε (t)kL∞ ≤ kΩ0ε kL∞ e
t
e
t
∞,1
+t kΩ0ε /rkL3,1 e
k div vε k 1 ∞ Lt L
o
.
6. Sub-critical regularities The main goal of this section is to prove Theorem 1, however we shall limit our analysis to the estimate of the lifespan of the solutions and to the rigorous justification of the low Mach number limit. For example we will not deal with the construction of maximal solution which is cassical and was done for instance in [18]. 6.1. Lower bound of Tε . We will show how the combination of Strichartz estimates with the special structure of axisymmetric flows allow to improve the estimate of the time lifespan Tε in the case of ill-prepared initial data. We shall prove the following result. Proposition 8. Let s >
5 2
and assume that sup k(v0,ε , c0,ε )kH s < +∞. ε≤1
Then the system (1.1) admits a unique solution (vε , cε ) ∈ C([0, Tε ]; H s ) such that Tε ≥ C0 log log log(ε−1 ). Moreover, there exists σ > 0 such that for every T ∈ [0, Tε ] ≤ C0 εσ + k∇cε kL1 B∞,1 k div vε kL1 B∞,1 0 0 T
T
and kΩε (T )kL∞ ≤ C0 eC0 T ,
k∇vε kL1 L∞ ≤ C0 eexp{C0 T } , T
k(vε , cε )(T )kH s ≤ C0 ee 22
exp{C0 T }
.
Proof. According to Proposition (7) we have Ckvε kL1 L2
kΩε (t)kL∞ ≤ kΩ0ε kL∞ e
t
n
C k div vε kL1 B0
e
t
∞,1
+t kΩ0ε /rkL3,1 e
k div vε k 1 ∞ Lt L
o
.
To bound kΩ0ε /rkL3,1 we use the fact that Ω0ε (0, 0, z) = 0, see Propostion 4, combined with Taylor formula Z 1 0 x1 ∂x1 Ω0ε (τ x1 , τ x2 , z) + x2 ∂x2 Ω0ε (τ x1 , τ x2 , z) dτ. Ωε (x1 , x2 , z) = 0
Applying (2.6) with a homogeneity argument yields Z 1 k∇Ω0ε (τ ·, τ ·, ·)kL3,1 dτ kΩ0ε /rkL3,1 . 0 Z 1 2 0 τ − 3 dτ . k∇Ωε kL3,1 0
. k∇Ω0ε kL3,1 .
According to the embedding H s−2 ֒→ L3,1 , for s > kΩ0ε /rkL3,1
From Proposition 1 we get
5 2
one has
. k∇Ω0ε kH s−2 . kΩ0ε kH s−1 . kv0,ε kH s .
C0 (1+t) exp k div vε kL1 B0
(6.1)
kΩε (t)kL∞ ≤ C0 e
Using Lemma 5 and Proposition 1 we get
t
∞,1
.
+ C0 kΩε (t)kL∞ 1 + Vε (t) . k∇vε (t)kL∞ . kvε (t)kL2 + k div vε (t)kB∞,1 0
Integrating in time and using Proposition 1 and Proposition 2, Z T kΩε (t)kL∞ 1 + Vε (t) dt + C0 k∇vε kL1 L∞ . kvε kL1 L2 + k div vε kL1 B∞,1 0 T
T
T
k div vε kL1 L∞
≤ C0 T e
T
0
+ C0 αε (T )eCVε (T ) + C0
Z
0
with
T
kΩε (t)kL∞ 1 + Vε (t) dt,
2s−5
αε (T ) := C0 ε r(2s−3) (1 + T 2 ). Using again Proposition 2 we obtain Z T k div vε kL1 L∞ CVε (T ) t kΩε (t)kL∞ (1 + Vε (t)dt. + C0 Vε (T ) ≤ C0 αε (T )e + C0 T e 0
Thus Gronwall’s lemma combined with (6.1) and Proposition 2 yield RT k div vε kL1 L∞ t Vε (T ) ≤ C0 eC0 0 kΩε (t)kL∞ dt αε (T )eCVε (T ) + T e C0 (1+T ) exp k div vε k 1 0 L B k div vε kL1 L∞ T ∞,1 t ≤ ee αε (T )eCVε (T ) + T e C0 (1+T ) exp k div vε k 1 0 Lt B ∞,1 ≤ ee αε (T ) eCVε (T ) + 1 (6.2)
≤ ee
C0 (1+T ) exp
αε (T )eCVε (T )
.
23
We choose Tε such that (6.3)
1 C0 e (1+Tε ) . > Cee αε (Tε )
log
Then we claim that for every t ∈ [0, Tε ]
1 . αε (Tε ) 1 Indeed, we set ITε := t ∈ [0, Tε ]; CVε (t) ≤ log αε (T . First this set is nonempty since 0 ∈ ITε . ε) By the continuity of t 7→ Vε (t), the set ITε is closed and thus to prove that ITε coincides with [0, Tε ] it suffices to show that ITε is an open set. Let t ∈ ITε then using (6.2) and (6.4) we get
(6.4)
CVε (t) ≤ log
CVε (t) ≤ Cee
e
≤ Ce
≤ Cee
αε (t)eCVε (t)
C0 (1+t) exp
αε (t) } αε (Tε )
C0 (1+t) exp{
C0 e (1+t)
< log
1 . αε (Tε )
This proves that t is in the interior of ITε and thus ITε is an open set of [0, Tε ]. Consequently we conclude that ITε = [0, Tε ]. Now we choose precisely Tε such that C0 e(1 + Tε ) = log log log(ε−θ ),
with θ > 0,
then the assumption (6.3) is satisfied whenever 2s−5
ε r(2s−3) (1 + Tε2 ) < εCθ · This last condition is satisfied for small values of ε when Cθ < (6.2) we get for T ∈ [0, Tε ] Vε (T ) ≤ ee
(6.5)
C0 e(1+T )
2s−5 r(2s−3) .
Now inserting (6.4) into
.
Plugging this into the estimate of Proposition 2 we obtain for T ∈ [0, Tε ] k div vε kL1 B∞,1 + k∇cε kL1 B∞,1 ≤ C0 eCe 0 0 T
exp{C0 e(1+T )}
T
2s−5
≤ C ε r(2s−3) ≤ Cεσ ,
(6.6)
2s−5
ε r(2s−3)
−Cθ
with
σ=
2s − 5 − Cθ. r(2s − 3)
From Corollary 2 and by choosing θ sufficiently small we obtain for every r ∈]2, +∞[ kQvε kLrT L∞ + kcε kLrT L∞
1
≤ C0 ε r (1 + T )eCVε (t) 1
≤ C0 ε r −Cθ log log log(ε−θ ) ′
≤ C0 εσ ,
σ ′ > 0.
We point out that the use of H¨older inequality and the slow growth of Tε allow us to get the following: for every r ∈ [1, +∞[ (6.7)
kQvε kLrT L∞ + kcε kLrT L∞
Inserting (6.6) into (6.1) leads to kΩε (T )kL∞ ≤ C0 eC0 T . 24
′
≤ C0 εσ ,
σ ′ > 0.
To estimate the solutions of (1.1) in Sobolev norms we use Proposition 1 combined with the Lipschitz bound (6.5) exp{C0 T } k(vε , cε )(T )kH s ≤ C0 ee . 6.2. Incompressible limit. Our task now is to prove the second part of Theorem 1. More precisely we will establish the following result. Proposition 9. Let (v0,ε , c0,ε ) be a H s -bounded family of axisymmetric initial data with s > that is sup k(v0,ε , c0,ε )kH s < +∞.
5 2,
0 0, 2 = 0. lim kPvε − vkL∞ T L
ε→0
However, the compressible and acoustic parts tend to zero: for every r ∈ [1, +∞[, T > 0 lim k(Qvε , cε )kLrT L∞ = 0.
ε→0
Proof. The last assertion concerning the compressible and acoustic parts of the fluid is a direct consequence of (6.7). Now we intend to show that for every time T > 0 the family (Pvε )ε is a 2 Cauchy sequence in L∞ T L for small values of ε and this will allow us to prove strong convergence, when ε tends to zero, of the incompressible parts to a vector-valued function v which is a solution of the incompressible Euler equations. For this purpose, let ε > ε′ > 0 and set ζε,ε′ = vε − vε′ and wε,ε′ = Pvε − Pvε′ . Applying Leray’s projector P to the first equation of (1.1) yields to It follows that L2
∂t Pvε + P(vε · ∇vε ) = 0.
∂t wε,ε′ + P(vε · ∇ζε,ε′ ) + P(ζε,ε′ · ∇vǫ′ ) = 0.
Taking the inner product of this equation with wε,ε′ and using the fact that Leray’s projector is an involutive self-adjoint operator we get Z Z 1d 2 (ζε,ε′ · ∇vǫ′ )Pwε,ε′ dx (vε · ∇ζε,ε′ )Pwε,ε′ dx − kwε,ε′ (t)kL2 = − 2 dt R3 R3 Z Z ((wε,ε′ + Qζε,ε′ ) · ∇vǫ′ )wε,ε′ dx (vε · ∇(wε,ε′ + Qζε,ε′ )) wε,ε′ dx − = − R3 R3 Z Z 1 = (wε,ε′ · ∇vǫ′ )wε,ε′ dx |wε,ε′ |2 div vε dx − 2 R3 R3 Z vε · ∇Qζε,ε′ + Qζε,ε′ · ∇vǫ′ wε,ε′ dx − R3 ≤ k div vε (t)kL∞ + k∇vε′ (t)kL∞ kwε,ε′ (t)k2L2 + kvε (t)kL2 k∇Qζε,ε′ (t)kL∞ + kQζε,ε′ (t)kL∞ k∇vε′ (t)kL2 kwε,ε′ (t)kL2 . Integrating in time and using Gronwall’s lemma
k div vε kL1 L∞ +k∇vε′ kL1 L∞ 0 t t kwε,ε′ (t)kL2 ≤ kwε,ε ′ kL2 + Fε,ε′ (t) e 25
with F
ε,ε′
(t) = kvε k
2 L∞ t L
k∇Qζ
ε,ε′
kL1t L∞ + kQζ
ε,ε′
Splitting Qζε,ε′ into low and high frequencies we get
kL1t L∞ k∇v k ε′
2 L∞ t L
.
0 + kQζε,ε′ kL1t L∞ . k∇Qζε,ε′ kL1t L∞ . kdiv ζε,ε′ kL1t B∞,1
Therefore combining Proposition 1 and Proposition 8 with (7.47) we get for some η > 0 sup Fε,ε′ (t) ≤ C0 εη .
t∈[0,Tε ]
According to Proposition 8 we obtain kwε,ε′ (t)kL2 ≤ C0 ee
exp{C0 t}
η 0 kwε,ε . ′ kL2 + ε
2 Thus we deduce that (Pvε )ε is a Caucy sequence in the Banach space L∞ T L and therefore it s ∞ converges strongly to some v which belongs to Lloc (R+ ; H ). This last claim is a consequence of the uniform bound in H s . It remains now to prove that v is a solution for the incompressible Euler system with initial data v 0 := limε→0 Pv0,ε . The passage to the limit in the linear part (∂t vε )ε is easy to do by using the convergence in the weak sense. For the nonlinear term we split the velocity into its compressible and incompressible parts:
vε · ∇vε = Pvε · ∇Pvε + Pvε · ∇Qvε + Qvε · ∇vε .
By virtue of Proposition 1, Proposition 8 and (6.7) we get kPvε · ∇Qvε kL1 L2 T
2 k∇Qvε kL1 L∞ ≤ kvε kL∞ T L T
≤ C0 kQvε kL1 L∞ + k div vε kL1 B∞,1 0 T
T
≤ C0 εη .
For the last term we combine (6.7) with Proposition 8, kQvε · ∇vε kL1 L2 T
It follows that
≤ kvε kL∞ 1 kQvε kL1 L∞ T H T
≤ C0 εη .
lim kvε · ∇vε − Pvε · ∇Pvε kL1 L2 = 0.
ε→0
T
The strong convergence of Pvε to v in
2 L∞ T L
allows us to claim that
lim kPvε ⊗ Pvε − v ⊗ vkL1 L1 = 0.
ε→0
T
Consequently, the family P(vε · ∇vε ) converges strongly in some Banach space to P(v · ∇v) and thus we deduce that v satisfies the incompressible Euler system. 7. Critical regularities 5 2 In this section we will study the system (1.1) with initial data lying in the critical Besov spaces B2,1 . There are some additional difficulties compared to the subcritical case that we can summarize in two points. The first one concerns the Beal-Kato-Majda criterion which is out of use and the second one is related to the extension of the interpolation argument used in Proposition 2 to the critical regularity. We will start with a strong version of Theorem 2 in which we require the initial data 5
,Ψ
2 to be in a more regular space of type B2,1 . We are able to prove the desired result for any slowly growth of Ψ. Roughly speaking, to get the incompressible limit and quantify the lower bound of the lifespan, the function Ψ must only increase to infinity. The study of the case Ψ ≡ 1 which is the subject of the Theorem 2 will be deduced from the Corollary 1 and the Theorem 3.
26
5
,Ψ
2 -bounded family of axisymmetric initial Theorem 3. Let Ψ ∈ U∞ and {(v0,ε , c0,ε )0 0 such that for u ∈ Bp,1 n ∈ N we have
kun k
3 p Bp,1
n−1 ≤ C n−1 kukL ∞ kuk
3
p Bp,1
.
To estimate F (t) we will use the following composition law, see for instance [28] : for s ∈]0, 1[, p, r ∈ s and ψ a diffeomorphism then we have [1, ∞], f ∈ Bp,r s s 1 + k∇ψksL∞ . kf ◦ ψkBp,r ≤ Ckf kBp,r 30
Applying this result combined with (7.6) we get for p ∈]3, ∞[ Z t 3 kdiv u(τ )k p3 1 + k∇ ψ(τ, ψ −1 (t, ·)) kLp ∞ dτ kF (t)k 3p ≤ Bp,1
Bp,1
0
.
Z
t
0
kdiv u(τ )k
3 p Bp,1
Rt
e
τ
k∇u(s)kL∞ ds
dτ.
Combining this estimate with (7.13) yields to Z t
Ck∇ukL1 L∞ F (t)
t kdiv u(τ )k (7.14) ke − 1 p3 ≤ Ce Bp,1
0
3
p Bp,1
dτ.
Putting together this estimate, (7.12) and (7.10) Ck∇ukL1 L∞ F (t) ˜ ˜ t kdiv uk . kΩq (t)k − 21 1 + e ke Ωq (t)k − 12 B∞,∞
B∞,∞
. k∆q Ω0 k
−1 2 B∞,∞
≤ Ck∆q Ω0 k
3 p L1t Bp,1
Ck∇ukL1 L∞ t kdiv uk eCVp (t) 1 + e
3 p L1t Bp,1
−1
2 B∞,∞
eCVp (t) .
This completes the proof of (7.5) in the case of the negative regularity. It remains to show the 1 2 estimate for B∞,∞ which is the hard part of the proof. But before giving precise discussions about the difficulties, we shall rewrite under a suitable way the stretching term of the equation (7.2). For this purpose we use Proposition 4-(3) leading to (∆q Ω0 )× eθ = 0. Now according to the Proposition ˜ q (t) × eθ = 0, and furthermore 6-(2) this geometric property is conserved through the time, that is Ω the equation (7.2) becomes ˜ ˜ q + (u · ∇)Ω ˜q + Ω ˜ q div u = ur Ω ∂t Ω r q (7.15) ˜ Ωq|t=0 = ∆q Ω0 .
Applying the maximum principle and using Gronwall’s inequality we obtain k(div u,ur /r)k
Lt L ˜ q (t)kL∞ ≤ k∆q Ω0 kL∞ e . kΩ ˜ q (t) × eθ = 0 we see that the solution Ω ˜ q has two components From the geometric constraint Ω ˜ q = (Ω ˜ 1q , Ω ˜ 2q , 0), and the mathematical analysis will be the same for both in the Cartesian basis, Ω ˜ 1. components. Thus we will just focus on the treatment of the first component Ω
(7.16)
ur r
u1 x1
1 ∞
q
u2 x2 ,
From the identity = = which is an easy consequence of ˜ 1q satisfies the equation function Ω ( ˜1 ˜ 1q + (u · ∇)Ω ˜ 1q + Ω ˜ 1q div u = u2 Ωq , ∂t Ω x2 ˜ 1q |t=0 = ∆q Ω1 . Ω 0
uθ
= 0, it is clear that the
Now we shall discuss the most difficulties encountered when we would like to propagate positive regularities and explain how to circumvent them. In the work of [1], the velocity is divergences free and thus the persistence of the the regularity B∞,∞ , 0 < s ≤ 1 is guaranteed by the special r v ˜ structure of the stretching term r Ωq where the axisymmetry of the velocity plays a central role. ˜ q div u does not in general belong In our case the velocity is not incompressible and the new term Ω s to any space of type B∞,∞ , s > 0 because in the context of critical regularities the quantity div u is only allowed to be in L∞ or in any Banach space with the same scaling. Our idea is to filtrate the compressible part leading to a new function governed by an equation where all the terms have positive regularity. But we need to be careful with the filtration procedure, its commutation with 31
the transport part leads to a bad term and that’s why we have to extract the compressible part after using the Lagrangian coordinates. For the simplicity, we will use the following notation: we denote by t 7→ x1 (t), x2 (t), x3 (t) = ψ(t, x), the path of the individual particle located initially at the position x. Now, we introduce the new functions: ˜ q (t, ψ(t, x)), ηq (t, x) = eG(t,x) Ω and G(t, x) =
Z
t
(div u)(τ, ψ(τ, x))dτ,
U (t, x) = u(t, ψ(t, x)).
0
˜ q , the second one As before, we shall restrict attention to the treatment of the first component of Ω 1 can be teated in a similar way. It is not hard to see that the function ηq satisfies the equation ∂t ηq1 (t, x) = U 2 (t, x)
(7.17)
ηq1 (t, x) , ηq (0, x) = ∆q Ω10 (x). x2 (t) η1 (t,x)
Accordingly, we deduce from this equation that the quantity t 7→ xq 2 (t) is conserved. Indeed, differentiate this function with respect to t, then use the equation of the flow we find ∂t
ηq1 (t, x) = x2 (t)
x2 (t)∂t ηq1 (t, x) − ηq1 (t, x)x′2 (t) x22 (t)
U 2 (t, x)ηq1 (t, x) − ηq1 (t, x)u2 (t, ψ(t, x)) x22 (t) = 0.
=
It follows that ηq1 (t, x) ∆q Ω10 (x) = · x2 (t) x2
(7.18)
1
2 Now, we are going to estimate ηq1 in the Besov space B∞,∞ and thus we prove the remaining case of (7.5) concerning the positive regularity. Applying Taylor’s formula to the equation (7.17) and using (7.18) we obtain
kηq1 (t)k 12 B∞,∞ (7.19)
Z t ηq1 (τ )
2 ≤ +
21 dτ
U (τ ) x2 (τ ) B∞,∞ 0 Z t
1
2 ηq (τ ) 1 ≤ k∆q Ω0 k 12 +
12 dτ.
U (τ ) x2 (τ ) B∞,∞ B∞,∞ 0 kηq1 (0)k 21 B∞,∞
To estimate the integrand in the right-hand side in the preceding estimate, we shall make use of the paradifferential calculus throughout Bony’s decomposition. So, we get by definition and from the triangular inequality,
1 ηq1 ηq1
2 ηq
2 2 1 U 1 1 ≤ + + 1 2 U ,
21 .
2
2
2
U
T ηq
TU
R x2 (t) B∞,∞ x (t) x (t) B∞,∞ B B 2 2 ∞,∞ ∞,∞ x2 (t)
The estimate of the first paraproduct and the remainder term are easy compared to the second paraproduct for which we will use some sophisticated analysis. Now for the first paraproduct we 32
write by definition and by the continuity of the Fourier cut-offs,
k X
k∆k (Sj−1 (ηq1 /x2 (t))∆j U 2 )kL∞ = sup 2 2
T ηq1 U 2 12 B∞,∞
x2 (t)
k≥−1
=
j∈N
sup 2
X
k 2
k≥−1
|j−k|≤4 j 2
k∆k (Sj−1 (ηq1 /x2 (t))∆j U 2 )kL∞
. sup 2 k∆j U 2 kL∞ kSj−1 (ηq1 /x2 (t)kL∞ j∈N
ηq1
x2 (t) L∞
j . sup 2j k∆j U 2 kL∞ sup 2− 2 Sj−1 j∈N
j∈N
η1
q . k∇U kL∞
− 12 , x2 (t) B∞,∞
(7.20)
where we have used Bernstein inequality in the last line. Now, using the identity (7.18) we get
∆ Ω1
q 0 =
− 21 x2 B∞,∞
η1
q
− 21
x2 (t) B∞,∞
1
. 2 2 q k∆q Ω0 kL∞ ≈ k∆q Ω0 k
1
2 B∞,∞
The proof of last inequality can be done as follows: applying Proposition 4-(3) we get the identity ∆q Ω10 (x1 , 0, x3 ) = 0. This yields in view of Taylor’s expansion Z 1 1 ∆q Ω0 (x1 , x2 , x3 ) = x2 (∂x2 ∆q Ω10 )(x1 , τ x2 , x3 )dτ 0
and hence we obtain by Lemma 8 and Bernstein inequality, Z 1
∆ Ω1
q 0 k(∂2 ∆q Ω10 )(·, τ ·, ·)k − 12 dτ ≤
− 21
x2 B∞,∞ B∞,∞ 0 Z 1 1 ≤ Ck∂2 ∆q Ω10 k − 21 τ − 2 dτ ≤
(7.21)
B∞,∞ 1 Ck∆q Ω0 k 12 . B∞,∞
0
Therefore we get by (7.20)
T
1 ηq x2 (t)
U 2
1
2 B∞,∞
. k∇U kL∞ k∆q Ω0 k
1
2 B∞,∞
.
Now according to Leibniz formula and (7.6) we find k∇U (t)kL∞ (7.22) Hence it follows that
T
(7.23)
≤ k∇u(t)kL∞ k∇ψ(t)kL∞ k∇ukL1 L∞
≤ k∇u(t)kL∞ e
1 ηq x2 (·)
U 2
1
2 L1t B∞,∞
t
k∇ukL1 L∞
≤ Ck∇ukL1t L∞ e Ck∇ukL1 L∞
≤ Ce
t
t
.
k∆q Ω0 k
k∆q Ω0 k
1
2 B∞,∞
33
1
2 B∞,∞
.
To estimate the remainder term we use its definition combined with (7.21),
ηq1
R U 2 ,
21 x2 (t) B∞,∞ (7.24)
.
sup
X
k≥−1 j≥k−3
1
k
e ηq 2 2 k∆j U 2 kL∞ ∆
j x 2 L∞
η1
q . kU kB∞,∞ 1
− 21 x2 (t) B∞,∞ k∆q Ω0 k 12 . . kU kB∞,∞ 1 B∞,∞
1 Straightforward computations combined with the embedding Lip(R3 ) ֒→ B∞,∞ imply that
kU (t)kB∞,∞ 1
. kU (t)kL∞ + k∇U (t)kL∞
. ku(t)kL∞ + k∇u(t)kL∞ k∇ψ(t)k∞
k∇ukL1 L∞
. ku(t)kL∞ + k∇u(t)kL∞ e kukL1 Lip
. ku(t)kLip e
t
t
.
Plugging this estimate into (7.24) yields to
ηq1
2 1
R U ,
2 x2 (·) L1t B∞,∞
(7.25)
CkukL1 Lip
≤ Ce
t
k∆q Ω0 k
1
2 B∞,∞
.
Now let us examine the second paraproduct which is more subtle and requires some special properties of the axisymmetric vector field u. First we write by definition
ηq1 ηq1 j
2 2
.
TU 2 21 . sup 2 Sj−1 U ∆j x2 B∞,∞ j∈N x2 (t) L∞ η1
q At first sight the term x2 (t) consumes a derivative more than what we require and the best configuration is to carry this derivative to the velocity U 2 . To do this we will use the commutation structure between the frequency cut-off operator ∆j and the singular multiplication by x12 . According to the identity (7.18) we write
Sj−1 U 2 (x)∆j
ηq1 (t, x) x2 (t)
∆q Ω10 x2 h 2 Sj−1 U (x) 1i = (∆q Ω10 ) ∆j ∆q Ω10 + Sj−1 U 2 (x) ∆j , x2 x2 := Ij (t, x) + IIj (t, x). =
Sj−1 U 2 (x)∆j
The estimate of the first term can be done as follows
S U 2 (t, x)
j−1
∞ kIj (t)kL .
∞ k∆j ∆q Ω10 kL∞ . x2 L
Using Proposition 5-(4) we get Sj−1 U 2 (x1 , 0, x3 ) = 0 and thus by Taylor’s formula and (7.22) we obtain
S U 2 (t, x)
j−1
∞ . k∇Sj−1 U 2 kL∞
x2 L . k∇U 2 kL∞ (7.26)
Ck∇ukL1 L∞
. k∇ukL∞ e 34
t
.
Since ∆j ∆q ω0 = 0 for |j − q| ≥ 2 then we deduce (7.27)
Ck∇ukL1 L∞
j
sup 2 2 kIj (t)kL∞ ≤ Ck∇u(t)kL∞ e
t
k∆q Ω0 k
1
2 B∞,∞
j
.
For the commutator term IIj we write by definition Z Sj−1 U 2 (t, x) 3j ∆q Ω10 (y) IIj (t, x) = 2 h(2j (x − y))(x2 − y2 ) dy x2 y2 R3 Sj−1 U 2 (x) 3j ˜ j ∆q Ω10 = 2−j 2 h(2 ·) ⋆ ( )(x) x2 y2 ˜ with h(x) = x2 h(x). Now we claim that for every f ∈ S ′ we have ˜ j ·) ⋆ f = 23j h(2
X
|j−k|≤1
˜ j ·) ⋆ ∆k f. 23j h(2
b b ˜ ˜ ⊂ supp ϕ. So we get 23j h(2 ˜ j ·) ⋆ Indeed, we have h(ξ) = i∂ξ2 b h(ξ) = i∂ξ2 ϕ(ξ). It follows that supp h ∆k f = 0, for |j − k| ≥ 2. This leads in view of Young inequality, (7.26) and (7.21) to
S U2 X j k ∆q Ω10
j−1 2− 2 k∆k ( )kL∞ sup 2 2 kIIj kL∞ . sup
∞ x2 x2 L j∈N j∈N |j−k|≤1
(7.28)
∆ Ω1
q 0 . k∇U kL∞
− 12 x2 B∞,∞ k∇ukL1 L∞
. k∇u(t)kL∞ e
t
k∆q Ω0 k
1
2 B∞,∞
.
Putting together (7.27) and (7.28) yields to
ηq1 Ck∇ukL1 L∞
t (7.29) k∆q Ω10 k 21 .
TU 2
1 21 ≤ Ce x2 (·) Lt B∞,∞ B∞,∞
Finally, combining (7.23),(7.25) and (7.29) we get
1 CkukL1 Lip
2 ηq t k∆q Ω10 k 12 .
U
1 12 . Ce x2 (·) Lt B∞,∞ B∞,∞ Inserting this estimate into (7.19) we obtain (7.30)
kηq1 (t)k
1 2 B∞,∞
CkukL1 Lip
≤ Ce
t
k∆q Ω10 k
1
2 B∞,∞
.
Next, we shall show how to derive the estimate (7.5) from the preceding one. First, we observe that ˜ q (t, x) = ηq (t, ψ −1 (t, x)). eF (t,x) Ω This will be combined with the following classical composition law 1
−1 2
f ◦ ψ −1 1 ≤ Ckf k 1 1 + k∇ψ k L∞ 2 2 B∞,∞
B∞,∞
leading acording to (7.6) and (7.30) to
kηq1 (t, ψ −1 (t, ·))k
1 2 B∞,∞
≤ Ckηq1 k
1 2 B∞,∞
≤ Ck∆q Ω10 k
Ck∇ukL1 L∞
e
1 2 B∞,∞
35
t
CkukL1 Lip
e
t
.
This achieves the proof of (7.5). According to (7.4), its local version reads as follows 1
˜ q (t))kL∞ ≤ C2− 2 |j−q|k∆q Ω0 kL∞ eCVp (t) . k∆j (eF (t) Ω
(7.31)
On the other hand, coming back to the equation (7.17), taking the L∞ -estimate and using Gronwall’s inequality we get ˜ q (t)kL∞ keF (t) Ω
= kηq (t)kL∞
Rt
≤ k∆q Ω0 kL∞ e
0
k(U 2 (τ )/x2 (τ )kL∞ dτ
.
Now, since ψ is an homeomorphism and u2 /x2 = ur /r then k(U 2 (τ )/x2 (τ )kL∞ = k(ur /r)(τ )kL∞ and thus we obtain r
F (t) ˜
ke
(7.32)
Ωq (t)k
≤ k∆q Ω0 k
L∞
L∞
k ur kL1 L∞
e
t
.
Set ζ(t, x) = eF (t,x) Ω(t, x), then by the linearity of the problem (7.2) we get the decomposition X ˜ q (t, x). ζ(t, x) = eF (t,x) Ω q≥−1
Let N ∈ N be an integer that will be fixed later, then using (7.31) and (7.32) we find X ˜ q (t) kL∞ kζ(t)kB∞,1 ≤ k∆j eF (t) Ω 0 j,q≥−1
≤
X
|j−q|≤N
˜ q (t) kL∞ + k∆j eF (t) Ω kur /rkL1 L∞ t
. N kΩ0 kB∞,1 e 0 Now we choose N ≈ Vp (t) then (7.33)
kur /rkL1 L∞
kζ(t)kB∞,1 ≤ CkΩ0 kB∞,1 e 0 0
t
X
|j−q|>N
− 21 N
+2
˜ q (t)kL∞ k∆j eF (t) Ω
kΩ0 kB∞,1 eCVp (t) . 0
1 + kdiv uk
3 p L1t Bp,1
+
Z
t 0
ku(τ )kLip dτ .
Our next object is to obtain an estimate for kΩ(t)kB∞,1 0 form ζ. For this purpose we develop similar calculous to (7.12) and (7.13). We start with the obvious identity Ω(t, x) = ζ(t, x) + (e−F (t,x) − 1)ζ(t, x), Using the law products: for p ∈ [1, ∞[ kuvkB∞,1 0 ≤ CkukB∞,1 0 kvk
3
p Bp,1
we obtain
,
kΩ(t)kB∞,1 . kζ(t)kB∞,1 1 + ke−F (t) − 1k 0 0
3 p Bp,1
Similarly to (7.14) we get for p ∈]3, +∞[ ke−F (t) − 1k
3 p Bp,1
Ck∇ukL1 L∞
≤ Ce
t
Z
0
36
.
t
kdiv u(τ )k
3
p Bp,1
dτ.
Combining these estimates with (7.33) we find Z t Ck∇ukL1 L∞ t kdiv u(τ )k kΩ(t)kB∞,1 . kζ(t)kB∞,1 1+e 0 0
3 p Bp,1
0
kur /rkL1 L∞ t
≤ CkΩ0 kB∞,1 e 0
dτ
Ck∇ukL1 L∞ t kdiv uk2 1+e
3
p L1t Bp,1
This achieves the proof of Theorem 4.
Z t ku(τ )kLip dτ . 1+ 0
7.3. Lower bound of Tε . The object of this section is to give the proof of Theorem 3 but for the sake of the conciseness we will restrict the attention only to the lower bound of Tε . At the same time we will derive the involved Strichartz estimates which form the cornerstone for the proof of the low Mach number limit and to avoid the redundancy we will omit the proof of this latter point which can be done analogously to the subcritical case. Proof. Using Lemma 5 we get + kΩε (t)kB∞,1 . kvε (t)kLip . kvε (t)kL2 + k div vε (t)kB∞,1 0 0 Integrating in time, using Proposition 1 and Proposition 2 we find + kΩε kL1 B∞,1 kvε kL1 Lip . kvε kL1 L2 + k div vε kL1 B∞,1 0 0 T
T
T
T
k div vε kL1 B0
≤ C0 (1 + T ) e
∞,1
t
. + kΩε kL1 B∞,1 0 T
Let Vε (T ) := kvε kL1 Lip + k∇cε kL1 L∞ T
T
then using again Proposition 2 we obtain (7.34)
k(div vε ,∇cε )kL1 B0
Vε (T ) ≤ C0 (1 + T )e
∞,1
t
. + kΩε kL1 B∞,1 0 T
Now, according to Theorem 4 we have (7.35)
kvεr /rkL1 L∞
kΩε (t)kB∞,1 ≤ C0 e 0
t
1 + eCVε (T ) kdiv vε k2
3
p L1t Bp,1
By virtue of Proposition 7 and Proposition 1, we get kvεr /rkL1t L∞ (7.36)
1+
Z
t 0
kvε (τ )kLip dτ .
k div vε kL1 B0
+ C0 T e ≤ Ckvε kL1 L2 + k div vε kL1 B∞,1 0
T
∞,1
T
T
Ck div vε kL1 B0
≤ C0 (1 + T ) e
T
∞,1
In the inequality (7.35) we need to estimate kdiv vε k
3
.
p L1t Bp,1
. For this purpose we will interpolate
this norm between the energy estimate and the dispersive one. More precisely, for p ∈]2, +∞[ we get by interpolation 2
kdiv vε k
3 p Bp,1
1− 2
≤ Ckdiv vε k p 3 kdiv vε kB 0 p . 2 B2,1
∞,1
Integrating in time and combining H¨older inequality with Proposition 1 we find kdiv vε k
3 p L1T Bp,1
(7.37)
1− 2
2
2
≤ CT p kdiv vε k p
3
2 L∞ T B2,1
p kdiv vε kL1 B 0 T
1− 2
2
p ≤ C0 T p eCVε (T ) kdiv vε kL1 B . 0 T
37
∞,1
∞,1
Putting together (7.35), (7.36) and (7.37) and taking p = 4 yield to Ck div vε kL1 B0 T ∞,1 0 1 + eCVε (T ) k div vε kL1 B∞,1 1 + Vε (T ) kΩε (t)kB∞,1 0 ≤ C0 exp C0 (1 + T ) e T
CeCVε (T ) k div vε kL1 B0 T ∞,1 ≤ C0 exp C0 (1 + T ) e 1 + Vε (T ) .
Combining this estimate with (7.34) and applying Gronwall’s inequality we obtain CeCVε (T ) k(div vε ,∇cε )k 1 0 L B∞,1 T exp C0 (1+T ) e (7.38) Vε (T ) ≤ C0 e .
Let N ∈ N∗ be an integer that will be fixed later. Using the dyadic decomposition combined with Bernstein inequality and the property (1) in the Definition 1 we get k div vε kB∞,1 0
= k div Qvε kB∞,1 0 X X k∆q div Qvε kL∞ = k∆q div Qvε kL∞ + q≥N
q 0.
≤ C0 εσ ,
Inserting (7.47) into (6.1) leads to kΩε (T )kL∞ ≤ C0 eC0 T . To estimate the solutions of (1.1) in Sobolev norms we use Proposition 1 combined with the Lipschitz bound (7.46) k(vε , cε )(T )k
5 ,Ψ 2 B2,1
≤ C0 ee
exp{C0 T }
.
8. Appendix This last section is devoted to the proof of some lemmata used in the preceding sections. Lemma 4. Let s >
5 2
and ϕ ∈ L∞ ([0, T ]; H s (R3 )), then we have ≤ CT k∇ϕkL1 B∞,1 0
2s−5 1− r(2s−3)
T
2s−5
2
2s−3 kϕkL2s−3 r L∞ kϕkL∞ H s . T
T
Proof. To prove this inequality we will P use a frequency interpolation argument. We split the function ϕ into its dyadic blocks: ϕ = q≥−1 ∆q ϕ and we take an integer N that will be be fixed later. Then by Bernstein inequality we obtain k∇ϕkB∞,1 0
=
N X
q=−1
k∆q ∇ϕkL∞ +
N
≤ C2 kϕkL∞ + C
X
k∆q ∇ϕkL∞
q>N
X
q>N
5
2 2 q k∆q ϕkL2 5
≤ C2 kϕkL∞ + C2−N (s− 2 ) kϕkH s . N
Now we choose N such that 3
2N (s− 2 ) ≈ and this gives
kϕkH s , kϕkL∞ 2
2s−5
2s−3 2s−3 k∇ϕkL∞ ≤ CkϕkH s kϕkL∞ .
Now to get the desired estimate, it suffices to use H¨older inequality in time variable.
The next lemma deals with a logarithmic estimate in the compressible context. Lemma 5. The following assertions hold true (1) Sub-critical case: Let v be a vector field belonging to H s (R3 ) with s > Ω = ∇ ∧ v its vorticity, then we have + kΩkL∞ log(e + kvkH s ). k∇vkL∞ . kvkL2 + kdiv vkB∞,1 0 1 ∩ L2 , then (2) Critical case: Let v be a vector field belonging to B∞,1
+ kΩkB∞,1 . k∇vkL∞ . kvkL2 + +kdiv vkB∞,1 0 0 40
5 2
and denote by
Proof. We split v into compressible and incompressible parts: v = Qv + Pv. Then curl v = curl Pv. Thus the incompressible part Pv has the same vorticity as the total velocity and thus we can use Brezis-Gallou¨et logarithmic estimate k∇PvkL∞
. kPvkL2 + kΩkL∞ log(e + kPvkH s ) . kvkL2 + kΩkL∞ log(e + kvkH s ).
It remains to estimate the Lipschitz norm of the compressible part Qv. Using Bernstein inequality for low frequency k∇QvkL∞
= k∇2 ∆−1 div vkL∞
≤ k∆−1 ∇2 ∆−1 div vkL∞ + . kvkL2 +
X q∈N
X q∈N
k∆q ∇2 ∆−1 div vkL∞
k∆q div vkL∞
. . kvkL2 + k div vkB∞,1 0 This concludes the proof of the logarithmic estimate.
We will establish the following Lemma 6. Let s > 0 and Ψ ∈ U , see the Definition 1. Then we have the following commutator estimate: X 2qs Ψ(q)k[∆q , v · ∇]ukL2 . k∇vkL∞ kukB s,Ψ + k∇ukL∞ kvkB s,Ψ . 2,1
q≥−1
2,1
Proof. By using Bony’s decomposition, we can split the commutator into three parts, [∆q , v · ∇]u = [∆q , Tv · ∇]u + [∆q , T∇· · v]u + [∆q , R(v, ∇)]u = Iq + IIq + IIIq . We start with the estimate of the first term I. Then by definition X k[∆q , Sj−1 v · ∇]∆j ukL2 . k[∆q , Tv · ∇]ukL2 ≤ |j−q|≤4
We can now use the following commutator inequality, for a proof see for example [9], k[∆q , a]bkLp ≤ C2−q k∇akL∞ kakLp which yields in view of Bernstein inequality to X 2−q k∇Sj−1 vkL∞ k∇∆j ukL2 k[∆q , Tv · ∇]ukL2 ≤ C |j−q|≤4
. Ck∇vkL∞
X
|j−q|≤4
. k∇vkL∞ 41
X
|j−q|≤4
2j−q k∆j ukL2
k∆j ukL2 .
Multiplying this last inequality by 2qs Ψ(q), summing up over q, and using the property (2) in the Definition 1 we get X Ψ(q) X 2qs Ψ(q)kIq kL2 . k∇vkL∞ 2js Ψ(j)k∆j ukL2 Ψ(j) q≥−1
|j−q|≤4
. k∇vk
L∞
kukB s,Ψ . 2,1
Similarly, the second term IIq is estimated by X 2qs Ψ(q)kIIq kL2 . k∇ukL∞ kvkB s,Ψ . 2,1
q≥−1
What is left is to estimate the remainder term. From the definition and since the Fourier transform of [∆q , ∆j v]∇ is supported in a ball of radius 2q then we obtain X e j ukL2 . k[∆q , R(v, ∇)]ukL2 ≤ k[∆q , ∆j v]∇∆ j≥q−4
To estimate the term inside the sum we do not need to use the structure of the commutator. Applying H¨older and Bernstein inequalities yields to X e j ukL∞ k∆j vkL2 k∇∆ k[∆q , R(v, ∇)]ukL2 . j≥q−4
. k∇ukL∞
2qs Ψ(q),
Multiplying this last inequality by property (1) of the Definition 1 we get X 2qs Ψ(q)k[∆q , R(v, ∇)]ukL2 q≥−1
X
j≥q−4
k∆j vkL2 .
summing up over q, using Fubini identity and the . k∇ukL∞ . k∇ukL∞
X X
2qs Ψ(q)k∆j ukL2
q≥−1 j≥q−4
X
k∆j ukL2
j≥−1
. k∇ukL∞ kukB s,Ψ .
X
2qs Ψ(q)
q≤j+4
2,1
This concludes the proof of Lemma 6.
The following proposition describes the propagation of Besov regularity for transport equation. Proposition 10. Let u be a smooth vector field, not necessary of zero divergence. Let f be a smooth solution of the transport equation ∂t f + u · ∇f = g, f|t=0 = f0 ,
s (R3 ) and g ∈ L1 (R ; B s ). Then the following assertions hold true such that f0 ∈ Bp,r + p,r loc (1) Let p, r ∈ [1, ∞]and s ∈]0, 1[, then Z t CV (t) s dτ , s s e−CV (τ ) kg(τ )kBp,r kf0 kBp,r + kf (t)kBp,r ≤ Ce 0
Rt
where V (t) = 0 k∇u(τ )kL∞ dτ and C is a constant depending on s. (2) Let s ∈] − 1, 0], r ∈ [1, +∞] and p ∈ [2, +∞] with s + 3p > 0, then Z t CV (t) p s dτ , s s e−CVp (τ ) kg(τ )kBp,r kf0 kBp,r + kf (t)kBp,r ≤ Ce 42
0
with Vp (t) = k∇ukL1t L∞ + kdiv uk
3
p L1t Bp,∞
.
Proof. (1) This estimate is classical, see for example [9] in the H¨olderian case and the similar proof works as well for Besov spaces. (2) The proof for a general case can be found for example in [10] and for the convenience of the reader we will give here the complete proof in our special case. We start with localizing in frequency the equation leading to, ∂t ∆q f + (u · ∇)∆q f = ∆q g + (u · ∇)∆q f − ∆q u · ∇f = ∆q g − [∆q , u · ∇]f.
Taking the Lp norm, then the zero divergence of the flow gives Z t Z t
k∆q gkLp dτ + (8.1) k∆q f (t)kLp ≤ k∆q f0 kLp +
[∆q , u · ∇]f p dτ. L
0
0
From Bony’s decomposition, the commutator may be decomposed as follows
[∆q , u · ∇]f = [∆q , Tu · ∇]f + [∆q , T∇· · u]f + [∆q , R(u, ∇)]f For the paraproducts we do not need any structure for the velocity, so reproducing the same the proof of Lemma 6 we find for s < 1 −qs s cq 2 , k[∆q , Tu · ∇]f + [∆q , T∇· · u]f kLp ≤ Ck∇ukL∞ kf kBp,r
with (cq )ℓr = 1. Concerning the remainder term we write X ˜ jf [∆q , R(u, ∇)]f = [∆q , ∆j ui ∂i ]∆ j
= ∂i
X
˜ jf − [∆q , ∆j ui ]∆
j≥q−4
X
˜ jf [∆q , ∆j div u]∆
j≥q−4
= Iq + IIq Similarly to the proof of Lemma 6 we get for s > −1
−qs s cq 2 kIq kLp ≤ Ck∇ukL∞ kf kBp,r ,
(cq )ℓr = 1
˜ j f is supported in a bull of size For the second term, since the Fourier transform of [∆q , ∆j div u]∆ q 2 then using Bernstein and H¨older inequalities we obtain for p ≥ 2 3
q ˜ jfk ≤ C2 p k[∆q , ∆j div u]∆
˜ j f kLp k[∆q , ∆j div u]∆
p
L2
3 q p
˜ j f kLp ≤ C2 k∆j div ukLp k∆ Therefore 2qs kIIq kLp
3
≤ C2 p q
X
j≥q−4
≤ Ckdiv vk
˜ j f kLp k∆j div ukLp k∆
3 p Bp,∞
Now since s +
3 p
X
j≥q−4
3
˜ j f kLp 2(q−j)(s+ p ) 2js k∆
> 0 then we can use Young inequality to deduce that s 2qs kIIq kLp r ≤ Ckdiv vk p3 kf kBp,r ℓ
Bp,∞
43
Combining these estimates yields for − min(1, 3p ) < s ≤ 0 to 2qs k[∆q , u · ∇]ukLp r ≤ Ckdiv vk
3
p Bp,∞
ℓ
s kf kBp,r
Plugging this estimate into (8.1) we get
s s kf (t)kBp,r . kf (0)kBp,r +
with
Z
t 0
s dτ + Vp (t)kf (τ )kBp,r
Vp (t) := k∇ukL∞ + kdiv uk
3
p Bp,∞
Z
0
t s dτ, kg(τ )kBp,r
.
To get the desired estimate it suffices to use Gronwall’s lemma. The last point that we will discuss concerns the action of the operator (∂r /r)∆−1 over axisymmetric functions. we will show that its restriction over this class of functions behaves like Riesz transforms. This study was done before in [17] and for the convenient of the reader we will give the complete proof here. Lemma 7. We have for every axisymmetric smooth scalar function u (∂r /r)∆−1 u(x) =
(8.2)
x21 x1 x2 x22 R u(x) + R22 u(x) − 2 2 R12 u(x), 11 2 2 r r r
with Rij = ∂ij ∆−1 . Proof. We set f = ∆−1 u, then we can show from Biot-Savart law that f is also axisymmetric. Hence we get by using polar coordinates that (8.3)
∂11 f + ∂22 f = (∂r /r)f + ∂rr f.
where ∂r =
x1 x2 ∂1 + ∂2 . r r
By using this expression of ∂r , we obtain ∂rr = =
x1 x2 2 x2 x2 x2 2x1 x2 x1 ∂1 + ∂2 = ∂r ( )∂1 + ∂r ( )∂2 + 21 ∂11 + 22 ∂22 + ∂12 . r r r r r r r2 x21 x22 2x1 x2 ∂ + ∂22 + ∂12 11 2 2 r r r2
since ∂r (
xi ) = 0, r
∀i ∈ {1, 2}.
This yields by using (8.3) that ∂r f r
x21 x22 2x1 x2 )∂ f + (1 − )∂22 f − ∂12 f 11 2 2 r r r2 x22 x21 2x1 x2 ∂ f + ∂22 f − ∂12 f. 11 2 2 r r r2
= (1 − =
To get (8.2), it suffices to replace f by ∆−1 u. The following result describes the anisotropic dilatation in Besov spaces. 44
s Lemma 8. Let s ∈] − 1, +∞[\{0}, f : R3 → R be a function belonging to B∞,∞ , and denote by fλ (x1 , x2 , x3 ) = f (λx1 , x2 , x3 ). Then, there exists an absolute constant C > 0 such that for all λ ∈]0, 1[ s s kfλ kB∞,∞ ≤ Cλs kf kB∞,∞ .
Proof. Let q ≥ −1, we denote by fq,λ = (∆q f )λ . From the definition we have s kfλ kB∞,∞
= k∆−1 fλ kL∞ + sup 2js k∆j fλ kL∞ . j∈N
For j, q ∈ N, the Fourier transform of ∆j fq,λ is supported in the set o n |ξ1 | + |ξ ′ | ≈ 2j and λ−1 |ξ1 | + |ξ ′ | ≈ 2q ,
where ξ ′ = (ξ2 , ξ3 ). A direct consideration shows that this set is empty if 2q . 2j or 2j−q . λ. Thus we get for an integer n1 X s 2js k∆j fq,λ kL∞ kfλ kB∞,∞ . k∆−1 fλ kL∞ + q−n1 +log2 λ≤j j≤q+n1
X
. k∆−1 fλ kL∞ +
q−n1 +log2 λ≤j j≤q+n1
s . k∆−1 fλ kL∞ + kf kB∞,∞
2(j−q)s 2qs kfq kL∞ n1 X
2js
−n1 +log2 λ s
s . k∆−1 fλ kL∞ + kf kB∞,∞ λ .
Let us now turn to the estimate of the low frequency k∆−1 fλ kL∞ . We observe that the Fourier transform of ∆−1 fq,λ vanishes when q ≥ n0 where n0 is an absolute integer. Consequently we get k∆−1 fλ kL∞
=
n0 X
k∆−1 fq,λ kL∞
q=−1 n0 X
≤ C
q=−1
k∆q f kL∞
s ≤ Ckf kB∞,∞
This completes the proof of the lemma.
Acknowledgments The author would like to thank J.-Y. Chemin, I. Gallagher and D. Li for the fruitful discussions about the critical case. References [1] H. Abidi, T. Hmidi and S. Keraani. On the global well-posedness for the axisymmetric Euler equations. Math. Ann. 347 (2010), no. 1, 15–41. [2] H. Abidi, T. Hmidi and S. Keraani. On the global regularity of axisymmetric Navier-Stokes-Boussinesq system. Discrete Contin. Dyn. Syst. 29 (2011), no. 3, 737–756, [3] T. Alazard. Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions. Adv. Differential Equations 10 (2005), no. 1, 19–44. [4] K. Asano. On the incompressible limit of the compressible Euler equation. Japan J. Appl. Math., 4 (1987), no. 3, 455–488. [5] H. Bahouri, J.Y. Chemin, R. Danchin. Fourier analysis and nonlinear partial differential equations. Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011. xvi+523 pp 45
[6] J. T. Beale, T. Kato, A. Majda, Remarks on the Breakdown of Smooth Solutions for the 3D Euler Equations, Comm. Math. Phys. 94 (1984) 61–66. [7] J. Bergh, J. L¨ ofstr¨ om, Interpolation spaces. An introduction, Springer-Verlag, 1976. [8] J.-M. Bony, Calcul symbolique et propagation des singularit´es pour les ´equations aux d´eriv´ees partielles non ´ lin´eaires, Ann. de l’Ecole Norm. Sup. 14 (1981) 209-246. [9] J.-Y. Chemin. Perfect incompressible fluids, volume 14 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York, 1998. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. [10] Q. Chen, C. Miao, and Z. Zhang. Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities. Rev. Mat. Iberoam. 26 (2010), no. 3, 915–946. ´ [11] R. Danchin. Zero Mach number limit in critical spaces for compressible Navier-Stokes equations. Ann. Sci. Ecole Norm. Sup. (4), 35 (2002), no. 1, 27–75. [12] B. Desjardins and E. Grenier. Low Mach number limit of viscous compressible flows in the whole space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455(1999), no. 1986, 2271–2279. [13] B. Desjardins, E. Grenier, P.-L. Lions, and N. Masmoudi. Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. (9), 78 (1999), no. 5, 461–471. [14] A. Dutrifoy and T. Hmidi. The incompressible limit of solutions of the two-dimensional compressible Euler system with degenerating initial data. Comm. Pure Appl. Math. 57 (2004), no. 9, 1159–1177. [15] I. Gallagher. R´esultats r´ecents sur la limite incompressible. S´eminaire Bourbaki. Vol. 2003/2004. Ast´erisque No. 299 (2005), Exp. No. 926, vii, 2957 [16] J. Ginibre and G. Velo. Generalized Strichartz inequalities for the wave equation. J. Funct. Anal., 133 (1995), no. 1, 50–68. [17] T. Hmidi and F. Rousset. Global well-posedness for the Euler-Boussinesq system with axisymmetric data. J. Funct. Anal. 260 (2011), no. 3, 745–796. [18] S. Klainerman and A. Majda. Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math., 34 (1981), no. 4, 481–524. [19] S. Klainerman and A. Majda. Compressible and incompressible fluids. Comm. Pure Appl. Math., 35 (1982), no. 5, 629–651. [20] O. A. Ladyzhenskaya, Unique solvability in large of a three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry. Zapisky Nauchnych Sem. LOMI 7 (1968), 155–177. [21] C.-K. Lin. On the incompressible limit of the slightly compressible viscous fluid flows. In Nonlinear waves (Sapporo, 1995), volume 10 of GAKUTO Internat. Ser. Math. Sci. Appl., pages 277–282. Gakk¯ otosho, Tokyo, 1997. [22] P.-L. Lions and N. Masmoudi. Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. (9), 77 (1998), no. 6, 585–627. [23] A. Majda. Compressible fluid flow and systems of conservation laws in several space variables, volume 53 of Applied Mathematical Sciences. Springer-Verlag, New York, 1984. [24] G. M´etivier and S. Schochet. The incompressible limit of the non-isentropic Euler equations. Arch. Ration. Mech. Anal., 158 (2001), no. 1, 61–90. [25] Rammaha, M. A. Formation of singularities in compressible fluids in two-space dimensions. Proc. Amer. Math. Soc. 107 (1989), no. 3, 705–714. [26] T. Shirota, T. Yanagisawa, Note on global existence for axially symmetric solutions of the Euler system, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), no. 10, 299–304. [27] Sideris, Thomas C. Formation of singularities in three-dimensional compressible fluids. Comm. Math. Phys. 101 (1985), no. 4, 475–485. [28] H. Triebel. Theory of function spaces, volume 78 of Monographs in Mathematics. Birkh¨ auser Verlag, Basel, 1983. [29] M. R. Ukhovskii, V. I. Yudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, Prikl. Mat. Meh. 32 (1968), no.1, 59–69. [30] S. Ukai. The incompressible limit and the initial layer of the compressible Euler equation. J. Math. Kyoto Univ., 26 (1086), no. 2, 323–331. IRMAR, Universit´ e de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex, France E-mail address:
[email protected]
46
