Navier-Stokes equations with variable density

Navier-Stokes equations with variable density Raphaël Danchin Abstract. We study the unique solvability of density-dependent incompressible Navier-Stokes equations in the whole space RN or in the torus TN (N ≥ 2). We aim at extending to this new context the celebrated results of wellN posedness in H˙ 2 −1 for Navier-Stokes equations with constant density (proved by H. Fujita and T. Kato in [13]). In the present paper, we show that local well-posedness holds for viscous fluids with non vanishing density. We further state global well-posedness in dimension N = 2. In large dimension, we show that global unique solutions do exist for small perturbations of an initial state with zero velocity and constant density. Our functional setting is very close to the one used by H. Fujita and T. Kato.

Introduction These last two decades, an increasing number of papers have been published on the so called Navier-Stokes equations   ∂t v + v · ∇v − µ∆v + ∇Π = 0, div v = 0, (0.1)  v|t=0 = v0 . Roughly, two main approaches have been used for solving the initial value problem associated to (0.1). The first one goes back to the pioneering work by J. Leray in 1934 (see [15]) and takes advantage of the conservation of energy, namely Z t def 2 2 2 (0.2) E(t) = ku(t)kL2 + 2µ k∇u(τ )kL2 dτ ≤ ku0 kL2 , 0

to build global (weak) solutions with finite energy. This approach is very satisfactory in dimension N = 2 as uniqueness may be proved. On the other hand, the question of uniqueness for finite energy solutions when N ≥ 3 has remained an open problem. A second kind of approach has been introduced by H. Fujita and T. Kato in the sixties (see e.g [13]). It is based on contractive mapping arguments and on the smoothing properties of the semi-group of the heat equation. That approach is particularly efficient in functional spaces which have the same scaling invariance as (0.1), in the sense that their norm is invariant for all ` > 0 by the transformation (0.3)

v0 (x) 7→ `v0 (`x),

v(t, x) 7→ `v(`2 t, `x).

1991 Mathematics Subject Classification. Primary 76D03; Secondary 35Q30. Key words and phrases. Viscous incompressible fluids, variable density, critical regularity. 1

2

¨ DANCHIN RAPHAEL

It leads to the following type of statement : “Typical statement”: Let E and F ⊂ C(R+ ; E) be two functional spaces whose norm is invariant by (0.3). For T > 0, denote by FT the local version of F pertaining to functions defined on [0, T ]. Under appropriate compatibility conditions on E and F , the following result holds true: for any data v0 ∈ E, there exists T > 0 such that (NSI) has a unique local solution v ∈ FT . If in addition kv0 kE µ, then that solution is global. In dimension N = 3, the first example of spaces (E, F ) for which the “typical statement” holds has been given by H. Fujita and T. Kato in [13]. There, E is the 1 homogeneous Sobolev space H˙ 2 and n o 1 1 1 F = u ∈ C(R+ ; H˙ 2 ) | t 4 ∇u ∈ C(R+ ; L2 ) and t 4 ∇u −→t→0 0 . It turns out that in dimension N = 2, the spaces E = L2 and F = L2 (R+ ; H˙ 1 ) are adapted so that two-dimensional Kato’s and Leray’s solutions are actually the same. Since then, (0.1) has been shown to be well-posed in a number of functional N p −1 spaces : Lebesgue spaces LN , Besov spaces B˙ p,r , and so on. Yet, the fundamental problem of proving the existence of global smooth solutions for any large smooth data has not been solved. Though exciting from a mathematical viewpoint, studying (0.1) is somewhat disconnected to applications in fluid mechanics. Indeed a “real fluid” is hardly homogeneous or incompressible. We here aim at investigating the robustness of the “typical statement” for incompressible fluids with variable density. A similar concern is also relevant for compressible fluids (see e.g [4]). The equations we are interested in read :  ∂t ρ + div ρu = 0,    ∂t (ρu) + div(ρu ⊗ u) − µ∆u + ∇Π = 0, (0.4) div u = 0,    (ρ, u)|t=0 = (ρ0 , u0 ), where ρ = ρ(t, x) ∈ R+ stands for the density, and u = u(t, x) ∈ RN , for the velocity field. The term ∇Π (the gradient of the pressure) may be seen as the Lagrange multiplier associated to the constraint div u = 0. The initial conditions (ρ0 , u0 ) are prescribed. Unless otherwise specified, we shall assume throughout that x belongs to the whole space RN or to the torus TN . Leray’s approach is still relevant for (0.4) : assuming that ρ0 ∈ L∞ is non√ negative and that ρ0 u0 ∈ L2 , one can prove the existence of global weak solutions (ρ, u) with finite energy : Z t √ √ 2 2 2 (0.5) k ρu(t)kL2 + 2µ k∇u(τ )kL2 dτ ≤ k ρ0 u0 kL2 . 0

In dimension N = 2 and under the additional assumption ∇u0 ∈ L2 , one can further get a pseudo conservation law involving the norm of ∇u in L∞ (0, T ; L2 ), and of ∂t u, ∇Π and ∇2 u in L2 (0, T ; L2 ). This provides smoother weak solutions. Even in this latter case however, the problem of uniqueness has not been solved. We refer to [1] and to [16] for an overview of results on weak solutions. Some recent improvements have been obtained by B. Desjardins in [10], [11] and [12].

NAVIER-STOKES EQUATIONS WITH VARIABLE DENSITY

3

Very few works address the question of unique solvability for (0.4). The main paper in this direction is probably the one written by O. Ladyzhenskaia and V. Solonnikov in the seventies (see [14]). There, the authors consider system (0.4) in a bounded domain Ω with homogeneous Dirichlet boundary conditions. Under the 2 assumption that u0 ∈ W 2−q ,q (q > N ) and ρ0 ∈ C 1 (Ω) is bounded away from zero, the results are the following : • In dimension N = 2, global well-posedness holds. • In dimension N = 3, local well-posedness holds. If in addition u0 is small in 2 W 2−q ,q , then global well-posedness holds. We aim at finding a framework as general as possible in which unique solvability holds true. We also would like this framework to be compatible with the “typical statement” in the case where ρ is a positive constant. Scaling considerations should help us to find an adapted functional framework. Remark that system (0.4) is invariant under the transformation (0.6)

(ρ0 (x), u0 (x)) 7→ (ρ0 (`x), ù0 (`x)), (ρ(t, x), u(t, x), Π(t, x)) 7→ (ρ(`2 t, `x), ù(`2 t, `x), `2 Π(`2 t, `x)).

Therefore choosing initial data (ρ0 , u0 ) such that ∇ρ0 and u0 belong to an adapted space E for the “ordinary” Navier-Stokes equations should give satisfactory results. Taking (ρ0 , u0 ) in W 1,N × (LN )N seems to be a possible choice. This should be compared with the spaces used by Ladyzhenskaia and Solonnikov. As a matter of fact, in a work in progress devoted to the case of bounded domains, we extend the results by Ladyzhenskaia and Solonnikov to such rough initial data (see [8]). In the present work, we consider fluids in RN or TN and use Sobolev spaces. Therefore, (0.6) suggests to choose initial data (ρ0 , u0 ) such that ∇ρ0 and u0 belong N to H˙ 2 −1 . As system (0.4) degenerates if ρ vanishes or becomes unbounded, we further assume that ρ±1 ∈ L∞ . We also suppose that ρ tends to some positive 0 constant (say 1) at infinity (or has average 1 in the periodic case). def

From now on, we make the change of unknown a = 1/ρ − 1, which leads to the following system :   ∂t a + u · ∇a = 0, ∂t u + u · ∇u − µ(1+a)∆u + (1+a)∇Π = 0, (0.7)  div u = 0.

For (at least) technical reasons however, we did not obtain well-posedness in the critical Sobolev spaces described above. Either the data have to be (slightly) more regular, or ρ0 , close to a constant. As a matter of fact, the critical regularity s (see the definition exponent may be achieved in the framework of Besov spaces B˙ 2,1 in section 1), and under a smallness hypothesis. Our first well-posedness result reads : Theorem 0.1. There exists a constant c depending only on N , and such that N N 2 −1 2 for all u0 ∈ B˙ 2,1 with div u0 = 0, and a0 ∈ B˙ 2,1 with ka0 k N2 ≤ c, there is a B˙ 2,1

T ∈ (0, ∞] such that system (0.7) has a unique solution (a, u, ∇Π) with N

N

N

2 −1 2 +1 u ∈ Cb ([0, T [; B˙ 2,1 ) ∩ L1 (0, T ; B˙ 2,1 ),

2 a ∈ Cb ([0, T [; B˙ 2,1 ),

If in addition ku0 k

N

−1 2 B˙ 2,1

≤ cµ, one can take T = +∞.

N

2 −1 ∇Π ∈ L1 (0, T ; B˙ 2,1 ).

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4 N

2 Remark 0.2. As B˙ 2,1 ,→ L∞ , we have a L∞ control on the density for free !

Remark 0.3. In dimension N = 2, the above result means that system (0.4) is “almost” well-posed for u0 ∈ L2 and ρ0 close to a constant in H 1 ∩L∞ , a result which may be compared with the assumptions needed for getting global weak solutions ` a la Leray. For more drastic perturbations of homogeneous fluids, we obtain the following result : N

Theorem 0.4. Let α ∈ (0, 1), β ∈ (0, 1 + α), a0 ∈ H 2 +α with 0 < b ≤ 1 + a0 , N and u0 ∈ H 2 −1+β with div u0 = 0. There exists a T > 0 such that system (0.7) has a unique solution (a, u, ∇Π) in the space N N N def e +α eT (H N2 +β−1 ) ∩ L e 1 (H N2 +β+1 ) e 1 (H N2 +β−1 ) . ETα,β = C )× C × L T (H 2 T T

Moreover, 0 < b ≤ 1 + a and the following energy equality is satisfied : Z t √ √ 2 2 2 (0.8) ∀t ∈ [0, T ], k( ρu)(t)kL2 + 2µ k∇u(τ )kL2 dτ = k ρ0 u0 kL2 . 0

eT (H s ) stands for a (large) subspace of C([0, T ]; H s ), and L e 1 (H σ ) is Above, C T 1 σ slightly larger than L (0, T ; H ). The reader is referred to definition 1.5 for more details. Remark 0.5. Higher orders of regularity may be considered. In dimension N = 2, the system with variable non-vanishing density is globally well-posed, a result which is in agreement with the homogeneous case and with those obtained in [14] in a bounded domain : Theorem 0.6. Assume that N = 2, α ∈ (0, 1) and β ∈ (0, 1 + α). Let a0 ∈ H 1+α with 0 < b ≤ 1 + a0 , and u0 ∈ H β with div u0 = 0. Then system (0.7) has a unique global solution (a, u, ∇Π) which belongs to ETα,β for all T > 0 and satisfies 0 < b ≤ 1 + a and the energy equality. Remark 0.7. For the sake of simplicity, we did not consider any external force f in the right-hand side of the momentum equation in (0.7). It is not hard to N 2 −1 extend theorem 0.1 to the case f ∈ L1 (0, T ; B˙ 2,1 ) whereas theorem 0.4 holds with e 1 (H N2 +β ). f ∈L T

Remark 0.8. In dimension N = 2, the situation is somewhat strange. On one hand, there exists global weak solutions (whose uniqueness has not been stated) for rather smooth initial velocity u0 ∈ H 1 but rough initial density (ρ0 ∈ L∞ only). On the other hand, we have just stated that unique global strong solutions do exist for rough velocity (u0 ∈ H β only) but smooth density (ρ0 ∈ H 1+α ). Open questions : • Even in the Besov spaces framework, we do not know how to get the critical regularity exponent if a0 is not small. • Global existence in R3 or T3 for small u0 but large a0 has still to be proved. Yet this qualitative result is known to be true in bounded domains (see [14]).


5

The paper is structured as follows. In the first section, we define some notation and functional spaces. Section 2 is devoted to the study of well-posedness under the assumption that a0 is small. In the third section, we remove the smallness assumption on a0 and sketch the proof of theorem 0.4. Global existence of strong solutions in dimension N = 2 is studied in the last section. 1. Definitions and notation Throughout the paper, C stands for a “harmless constant” whose precise meaning will be clear from the context. We shall sometimes alternatively use the notation A . B instead of A ≤ CB, and A ≈ B means that A . B and B . A. To define the Besov spaces that we shall use, and to localize system (0.7), a Littlewood-Paley decomposition is needed. An (homogeneous) Littlewood-Paley decomposition relies upon a dyadic partition of unity : let ϕ ∈ C0∞ (RN ) be supdef

ported in, say, C = {ξ ∈ RN , 3/4 ≤ |ξ| ≤ 8/3} and such that X ϕ(2−q ξ) = 1 if ξ 6= 0. q∈Z

Denoting h = F −1 ϕ, we then define the dyadic blocks as follows Z ˙ q u def ∆ = ϕ(2−q D)u = 2qN h(2q y)u(x − y) dy. RN

The formal decomposition (1.1)

u=

X

˙ qu ∆

q∈Z

holds true modulo polynomials : if u ∈ S 0 (RN ) then P[RN ] and (1.1) holds in S 0 (RN )/P[RN ]. One can now define homogeneous Besov spaces :

P

q∈Z

˙ q u converges modulo ∆

Definition 1.1. For s ∈ R, (p, r) ∈ [1, +∞]2 and u ∈ S 0 (RN ), we set X r1 def r kukB˙ s = 2rsq k∆q ukLp p,r

q∈Z

with the usual change if r = +∞, and we denote n o s B˙ p,r = u ∈ S | kukB˙ s < +∞ . p,r

s s For s < N/p (s ≤ N/p if r = 1), we then define B˙ p,r as the completion of B˙ p,r for k·kB˙ s . If m ∈ N and N/p+m ≤ s < N/p+m+1 (N/p+m < s ≤ N/p+m+1 if p,r s r = 1), B˙ p,r is defined as the subset of distributions u ∈ S 0 modulo a polynomial of degree m, and such that ∂ α u ∈ B˙ s−m whenever |α| = m. p,r

s The definition of B˙ p,r does not depend on the choice of the Littlewood-Paley. s We shall make an extensive use of spaces B˙ 2,2 which are nothing but the “usual” s s ˙ Sobolev spaces H , and of the slightly smaller spaces B˙ 2,1 which have some nice embedding and product properties. Let us state some classical properties for those Besov spaces.

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6

Proposition 1.2. The following properties hold: i) Derivatives: there exists a universal constant C such that s−1 ≤ Ckuk ˙ s . C −1 kukB˙ s ≤ k∇ukB˙ p,r B p,r

p,r

More generally, if A is an homogeneous PDO of degree m then s−m . kuk ˙ s . kA(D)ukB˙ p,r B p,r

1

1

s−N ( − ) ii) Sobolev embeddings: if p1 < p2 and r1 ≤ r2 then B˙ ps1 ,r1 ,→ B˙ p2 ,r2 p1 p2 . In particular, we have N

s s B˙ 2,1 ,→ H˙ s ,→ B˙ 2,∞

and

2 ,→ L∞ . B˙ 2,1 N

s 2 iii) Algebraic properties: for s > 0, B˙ p,r ∩ L∞ is an algebra. So does B˙ 2,1 . θs +(1−θ)s 1 2 s1 s2 ˙ ˙ ˙ iv) Complex interpolation: [Bp,r , Bp,r ]θ = Bp,r .

Let us now state some results of continuity in Besov spaces for the usual product (see [17], section 4.4, for more details) : Proposition 1.3. The following inequalities hold true: kuvkB˙ s . kukL∞ kvkB˙ s + kvkL∞ kukB˙ s

(1.2) (1.3) kuvk

2,1

N

s1+s2− 2 B˙ 2,1

2,1

. kukB˙ s1 kvkB˙ s2 2,1

2,1

if

2,1

s1 , s2 ≤ N/2

and

if

s > 0,

s1 +s2 > 0.

Remark 1.4. We shall make an extensive use of the estimate kuvk

N

−1 2 B˙ 2,1

. kuk

N

2 B˙ 2,1

kvk

N

−1 2 B˙ 2,1

.

For treating the case when the density may be a large perturbation of a constant, the usual nonhomogeneous Sobolev spaces H s will be needed. We just want to emphasize here that they may be alternately defined by mean of a nonhomogeneous Littlewood-Paley decomposition : ˙ q if q ≥ 0, ∆q = ∆ X ∆−1 = χ(D) with χ(ξ) = 1 − ϕ(2−q ξ). q≥0

Now, it is easy to check that Z 12 X 12 s def 2 2 2 2qs kukH s = 1 + |ξ| |b u(ξ)| dξ ≈ 2 k∆q ukL2 . q≥−1

For X Banach space, we denote by C([0, T [; X) the set of continuous functions on [0, T [ with values in X, and by Cb ([0, T [; X) the subset of bounded functions of C([0, T [; X). For p ∈ [1, +∞], the notation Lp (0, T ; X) stands for the set of measurable functions on [0, T [ with values in X, such that t 7→ kf (t)kX belongs to Lp (0, T ). s In the present paper, X will be a (homogeneous) Besov space B˙ 2,1 or a nonhos mogeneous Sobolev space H . In the former case, we will be induced to localize the equations through Littlewood-Paley decomposition, therefore ending up with L2 estimates for each dyadic block. Performing a time integration should be the next step, but in doing so, we obtain bounds in spaces which are not of type Lρ (0, T ; H˙ s ) (except if ρ = 2). That remark naturally leads to the following definition :


7

Definition 1.5. Let ρ ∈ [1, +∞], T ∈]0, +∞] and s ∈ R. We set X Z T ρ2 12 def ρ 2qs 2 k∆q u(t)kL2 dt kukLeρ (H s ) = . T

0

q≥−1

Let us stress that by virtue of Minkowski inequality, we have kukLeρ (H s ) ≤ kukLρ (H s ) T T kukLρ (H s ) ≤ kukLeρ (H s ) T

if if

T

ρ ≤ 2, ρ ≥ 2.

We will often use the following interpolation inequality: (1.4) 1 θ 1−θ = + , s = θs1 +(1−θ)s2 . ρ ρ1 ρ2 Let us state some estimates for the product in those spaces, the proof of which is a straightforward adaptation of the one for ordinary Sobolev spaces (see e.g [17]): θ

1−θ

kukLeρ (H s ) ≤ kukLeρ1 (H s1 ) kukLeρ2 (H s2 ) with T

T

T

Proposition 1.6. If s > 0 and 1/ρ2 + 1/ρ3 = 1/ρ1 + 1/ρ4 = 1/ρ ≤ 1 then kuvkLeρ (H s ) . kukLρ1 (L∞ ) kvkLeρ4 (H s ) + kvkLρ2 (L∞ ) kukLeρ3 (H s ) . T

T

T

T

T

If s1 , s2 < N/2, s1 + s2 > 0 and 1/ρ1 + 1/ρ2 = 1/ρ ≤ 1 then kuvk eρ

LT (H s1 +s2 −

N 2

)

. kukLeρ1 (H s1 ) kvkLeρ2 (H s2 ) . T

T

2. Small perturbation of an homogeneous fluid In this section, we give an insight into the proof of theorem 0.1. 2.1. A priori estimates for the solution. Introducing P the Leray projector on solenoidal vector fields, system (0.7) rewrites  ∂t a + u · ∇a = 0,    ∂t u − µ∆u = P a(µ∆u − ∇Π) − u · ∇u , (2.1)    ∆Π = div a(µ∆u − ∇Π) − div(u · ∇u).

As a is merely advected by the vector field u whose gradient belongs to L1 (0, T ; L∞ ) N N (∇u is expected to belong to L1 (0, T ; B˙ 2 ), and B˙ 2 ,→ L∞ ), the initial regularity 2,1

2,1

of a is preserved. More precisely, we have (see the proof in the appendix of [8]) : Proposition 2.1. Let s ∈ (−1−N/2, 1+N/2] and v be a solenoidal vector field N s s 2 such that ∇v belongs to L1 (0, T ; B˙ 2,1 ). Suppose that f0 ∈ B˙ 2,1 , F ∈ L1 (0, T ; B˙ 2,1 ) ∞ s 0 ˙ and that f ∈ L (0, T ; B2,1 ) ∩ C([0, T ]; S ) solves ∂t f + v · ∇f = F, (T ) f|t=0 = f0 . There exists a constant C depending only on s and N , and such that the following inequality holds true: Z t (2.2) kf (t)kB˙ s ≤ eCV (t) kf0 kB˙ s + e−CV (τ ) kF (τ )kB˙ s dτ 2,1

with V (t) =

Rt 0

k∇v(τ )k

2,1

N 2 B2,1

0

2,1

s dτ . Moreover, f belongs to C([0, T ]; B˙ 2,1 ).

¨ DANCHIN RAPHAEL

8

Therefore, we have Ck∇uk

kak

(2.3)

N ˙ 2 L∞ T (B2,1 )

≤e

N ˙ 2 ) L1 (B 2,1 T

ka0 k

N

2 B˙ 2,1

.

Now, two types of terms appear in the right-hand side of the remaining equations in (2.1). We do not have to worry about the quadratic term u · ∇u which is also present in the “ordinary” Navier-Stokes equations. On the other hand, taking a small enough should enable us to get rid of the “new term” a(µ∆u − ∇Π) (as it is going to be absorbed in the estimates). Of course, a functional framework has to be found in which those arguments are rigorous. For that aim, we shall use the following estimates for the heat equation (due to J.-Y. Chemin in [3]) : Proposition 2.2. Let s ∈ R and v0 be a divergence-free vector field with s−1 coefficients in B˙ 2,1 and f be a time dependent vector field with coefficients in s−1 s−1 s+1 1 L (0, T ; B˙ 2,1 ). Let v ∈ C([0, T [; B˙ 2,1 ) ∩ L1 (0, T ; B˙ 2,1 ) be a solution to the heat equation ∂t v − µ∆v = f, (2.4) v|t=0 = v0 . There exists c = c(N ) such that the following estimate holds: kvkL∞ (B˙ s−1 ) + cµkvkL1 (B˙ s+1 ) ≤ kv0 kB˙ s−1 + kf kL1 (B˙ s−1 ) . 2,1

T

2,1

T

2,1

2,1

T

Applying the proposition above to the second equation in (3.2) and using that P is an homogeneous PDO of degree 0 yields kuk

N −1

˙ 2 L∞ T (B2,1 )

+ µkuk

. ku0 k

N +1

2 L1T (B˙ 2,1 )

N

−1 2 B˙ 2,1

+ ku · ∇uk

N −1

2 L1T (B˙ 2,1 )

+ka(µ∆u − ∇Π)k

N −1

2 L1T (B˙ 2,1 )

.

By virtue of (1.3), we get kuk

N

˙ 2 −1 L∞ T (B2,1 )

+ µkuk

(2.5)

. ku0 k

N

+1 2 L1T (B˙ 2,1 )

+kak

N

−1 2 B˙ 2,1

N ˙ 2 L∞ T (B2,1 )

+ kuk

N

˙ 2 −1 ) L∞ T (B2,1

µk∆uk

N −1 2 L1T (B˙ 2,1 )

k∇uk

+ k∇Πk

N −1 2 L1T (B˙ 2,1 )

From the last equation in (2.1), we gather k∇Πk 1 N2 −1 . kak ∞ N2 µk∆uk 1 N2 −1 +k∇Πk LT (B˙ 2,1 )

LT (B˙ 2,1 )

LT (B˙ 2,1 )

N

2 ) L1T (B˙ 2,1

N −1 2 L1T (B˙ 2,1 )

+kuk

.

N −1

˙ 2 L∞ T (B2,1

k∇uk

N

.

k∇uk

N

.

)

2 ) L1T (B˙ 2,1

Adding that latter inequality to (2.5) yields kuk

N

˙ 2 −1 L∞ T (B2,1 )

+kak

+ µkuk 1 N2 +1 + k∇Πk 1 N2 −1 . ku0 k N2 −1 LT (B˙ 2,1 ) LT (B˙ 2,1 ) B˙ 2,1 N µkuk 1 N2 +1 + k∇Πk 1 N2 −1 + kuk ∞ 2

˙ L∞ T (B2,1 )

LT (B˙ 2,1 )

LT (B˙ 2,1 )

N −1

2 LT (B˙ 2,1

)

2 ) L1T (B˙ 2,1

Now, it is clear that if a0 has been chosen small enough, a will remains small on some non empty time interval (see (2.3). If in addition, u0 is also small, the last line in the above inequality may be absorbed by the left-hand side, which enables us to close the estimates globally in time.


9

A similar result holds on a bounded time interval even if u0 is not small. It is only a matter of splitting u into u = u + uL where uL is the free solution to the heat equation (2.4) with right-hand side 0 and initial datum u0 , then going along the same lines as for the case u small (see [6] for more details). Once the above estimates stated, the existence of solution may be easily proved by mean of a standard iterative scheme. 2.2. Remarks on uniqueness. The equations for def

(δa, δu, δΠ) = (a2 −a1 , u2 −u1 , Π2 −Π1 ) read

(2.6)

 ∂t δa + u2 · ∇δa = −δu · ∇a1 ,   

∂t δu + u2 · ∇δu − µ∆δu + ∇δΠ = −δu · ∇u1    +a1 (µ∆δu−∇δΠ) + δa(µ∆u2 −∇Π2 ).

Obviously, the right-hand side of the first equation is responsible for the loss of one derivative in the estimates involving δa. This induces us to bound (δa, δu, ∇δΠ) in the space N N N N N N 2 −1 2 2 −2 2 −2 C([0, T ]; B˙ 2,1 ) × L1T (B˙ 2,1 ) ∩ C([0, T ]; B˙ 2,1 ) × L1T (B˙ 2,1 ) . This loss of one derivative is not going to be to much a trouble in dimension N ≥ 3. In the two-dimensional case however, we are induced to use a (forbidden) endpoint estimate in some nonlinear terms in (2.6) : if we assume that δu 1 1 belongs to L1 (0, T ; B˙ 2,1 ) and a1 ∈ L∞ (0, T ; B˙ 2,1 ), the term a1 ∆δu does not be−1 1 long to L (0, T ; B˙ 2,1 ) (as required to close the estimates according to proposition −1 2.2) but rather to the larger space L1 (0, T ; B˙ 2,∞ ). The reason why is that the −1 −1 1 product is continuous from B˙ 2,1 × B˙ 2,1 to B˙ 2,∞ only. Therefore, the right-hand −1 side of the second equation in (2.6) may be estimated in L1 (0, T ; B˙ 2,∞ ) only. A straightforward generalization of proposition 2.2 yields bounds for δu in the (larger) −1 1 1 ) and ) ∩ L∞ (0, T ; B˙ 2,∞ ). Now, assuming a1 ∈ L∞ (0, T ; B˙ 2,1 space L1 (0, T ; B˙ 2,∞ −1 1 1 1 1 ˙ δu ∈ L (0, T ; B2,∞ ) suffices to get an estimate for a ∆δu in L (0, T ; B2,∞ ). 1 Yet, we still are in trouble for the bound for having δu in L1 (0, T ; B˙ 2,∞ ) does not yield appropriate estimates for the right-hand side of the first equation in (2.6). We lack an additional bound in L1 (0, T ; L∞ ) for δu. The key to that ultimate difficulty is given by some logarithmic interpolation inequality. Then a (generalized) Gronwall estimate yields the stability (see [6] for more details). 3. Local well-posedness in the general case In this section, we aim at proving theorem 0.4. Compare to the previous section, we have to get rid of the smallness condition. The price to pay for that is some extra N N regularity requirement on the data : a0 is in H 2 +α and H 2 −1+β for some small positive α and β. We expect to glean some smallness from that extra smoothness so that it will be possible to close estimates even though a may be large. Once again, as ∇u is expected to be (at least) in L1 (0, T ; L∞ ), there will be no difficulty coming from the transport equation for a. Indeed, slight changes to the

¨ DANCHIN RAPHAEL

10

proof of proposition 2.1 yield (see the proof in [9]) C

kak e∞

(3.1)

≤e

N +α 2 )

Lt (H

Rt 0

k∇v(τ )k

Note that this estimate is standard with kak

N H 2 ∩L∞

L∞ t (H

dτ

ka0 k

N +α 2 )

H

N +α 2

.

as a left-hand side. There-

fore, we are left with the momentum equation. 3.1. The linearized momentum equation. This section is devoted to the proof of estimates for the following linear system :   ∂t u + v · ∇u + b(∇Π − µ∆u) = f, div u = 0, (3.2)  u|t=0 = u0 , def

where b, f , v and u0 are given. We shall further assume that b = inf x b(x) > 0 and that b tends to some positive constant (that we can assume to be 1 with no loss of def

generality) at infinity. We shall denote a = b − 1. Proposition 3.1. Let α ∈ (0, 1), p ∈ [1, +∞), p0 conjugate exponent of p, def

def

and s ∈ (1, min(1, 2/p) + α + N/2). Denote α e = min(α, (s − 1)/2) and AT = 1+b−1 kak e∞ N2 +eα . There exists C = C(s, N, α, p) and a positive exponent κ = LT (H

)

κ(s, α) such that

CAκ

V (T )

kukLe∞ (H s−1 ) + µbkukLe1 (H s+1 ) + k∇ΠkLe1 (H s−1 ) ≤ CAκT e T (µb)p−1 T T T × ku0 kH s−1 + kf kLe1 (H s−1 ) + µbAT kukLe1 (H s−α+1 e ) . T

T

with def

V (T ) =

 Z T   k∇v(t)k N2 ∞ dt   H ∩L  0 Z     

0

T

p

k∇v(t)k

H

N− 2 2 p0

dt

if

if

p = 1,

p > 1.

Sketchy proof of proposition 3.1 in the case p = 1 : It lies on spectral localization of (3.2) by mean of the nonhomogeneous Littlewood-Paley decomposition defined in section 1. With no loss of generality, one can assume that s ∈ [1 + 2α, 1 + α + N/2). Indeed, if s ∈ (1, 1 + 2α), it is only a matter of changing α into (s − 1)/2. Applying ∆q to (3.2), we get ∂t ∆q u − µ div(b∆q ∇u) + v · ∇∆q u + ∆q ∇Π = ∆q f − ∆q (a∇Π) + [v, ∆q ] · ∇u + µRq with def

Rqj = ∆q (b∆uj ) − div(b∆q ∇uj ). Taking the L2 -scalar product with ∆q u and doing elementary computations yields 1 d 2 2 k∆q ukL2 + µb k∇∆q ukL2 ≤ k∆q ukL2 µ kRq kL2 2 dt + k[v, ∆q ] · ∇ukL2 + k∆q f kL2 + ∆q (a∇Π) | ∆q u .


11

By virtue of Bernstein inequality, we have k∆q ∇ukL2 ≥ k2q k∆q ukL2 for some universal constant k > 0 whenever q ≥ 0. Next, as s ∈ (1 − N/2, 1 + α + N/2), the commutators may be bounded according to the appendix of [7] : Z T X 12 2 22q(s−1) k[v, ∆q ] · ∇ukL1 (L2 ) . k∇v(t)k N2 ku(t)kH s−1 dt, ∞ T

X

B2,∞ ∩L

0

q≥−1

2

22q(s−1) k∆q (a div w)−div(a∆q w)kL1 (L2 ) T

12

q≥−1

. k∇ak e∞

LT (H

N +α−1 2 )

kwkLe1 (H s−α ) . T

Finally, the term ∆q (a∇Π) | ∆q u may be bounded by mean of elementary paradifferential calculus (namely Bony’s decomposition introduced in [2]) combined with integration by parts. Using also proposition 1.6, one ends up with kukLe∞ (H s−1 ) + µbkukLe1 (H s+1 ) . ku0 kH s−1 + kf kLe1 (H s−1 ) + µbkukLe1 (H s+1−α ) T T T ZT T +µkak e∞ N2 +α k∇ukLe1 (H s−α ) + k∇vk N2 ∞kukH s−1 dτ LT (H

)

T

H

0

(3.3)

+kak e∞

LT (H

N +α 2 )

∩L

k∇ΠkLe1 (H s−1−α ) . T

The pressure Π verifies an elliptic equation : div(v∇Π) = div F with def

F = f + µa∆u − v · ∇u.

(3.4) In [7], we prove that bk∇ΠkLe1 (H s−1−α ) . T

1+

k∇ak e∞

LT (H

N +α−1 2 )

b

s−1α−α

kQF kLe1 (H s−1−α ) , T

def

where Q = I − P denotes the projector on gradient vector fields. According to proposition 1.6, we have kQF kLe1 (H s−1−α ) . kQf kLe1 (H s−1−α ) T

T

+µkak e∞

LT (H

N +α 2 )

k∆ukLe1 (H s−1−α ) + T

Z

T

kukH s−1 k∇vk

H

0

N 2

∩L∞

N 2

∩L∞

dt.

Now, coming back to (3.3), one concludes that s−1

kukLe∞ (H s−1 ) + bµkukLe1 (H s+1 ) . ku0 kH s−1 + ATα kf kLe1 (H s−1 ) T T T Z T s−1+α s−1 +µbAT α kukLe1 (H s+1−α ) + ATα kukH s−1 k∇vk T

H

0

dt.

Gronwall lemma completes the proof of the desired inequality. 3.2. Outline of the proof of theorem 0.4. It suffices to show that the estimates may be closed for small enough time. Indeed, the existence will then stem from some standard iterative scheme, and uniqueness is not going to be a trouble (even in dimension N = 2) as we are given some extra smoothness compare to the previous section. According to inequality (3.1), a control on u in L1 (0, T ; H with a bound of the type kak e∞

LT (H

N +α 2 )

≤ M ka0 k

H

N +α 2

N 2

+1+β

) provides us

¨ DANCHIN RAPHAEL

12

for some M ≥ 1. Applying proposition 3.1 with p = 1 and s = β + N/2 to the momentum equation in (0.7), and using Sobolev embeddings yields def

U (T ) = kuk e∞

LT (H

N +β−1 2 )

+ µbkuk e1

LT (H

CAκ T kuk

≤ CAκT e

N +β+1 2 )

N e 1 (H 2 +β+1 ) L T

+ k∇Πk e1

ku0 k

LT (H

H

N +β−1 2

N +β−1 2 )

+ µbAT kuk e1

LT (H

For closing the estimates, we lack a bound for the last term. On the other hand, by interpolation, we have α2e kuk e1 kuk 2−2 αe N +β+1−αe ≤ kuk e∞ N2 +β−1 so that

e L T

(H

2

LT (H

)

µbkuk e1

LT (H

N +β+1−α e 2 )

LT (H

)

N +β+1 2 )

N +β+1−α e 2 )

.

1− α2e

α e

≤ (µbT ) 2 U (T ).

Now, it is clear that this term becomes negligible for small T , thus estimates may be closed on a small time interval. 4. Global well-posedness in dimension N = 2 Obviously, the proof used hitherto cannot yield global well-posedness even under some smallness condition on the data. The reason why is that a time dependence appears when bounding the last term in the estimate of proposition 3.1. A Poincaré inequality would enable us to avoid the appearance of that bad term. As a matter of fact, it was a key argument for proving global results in bounded two-dimensional domains (see [14]). In the whole space however, no inequality of this type is expected. More surprisingly, the situation is not better in the torus as we do not have a good control on the average of u ! On the other hand, one can observe that the missing control on the low frequencies of u is given for free by a pseudo-conservation law involving the H 1 norm of the velocity. At first sight, this seems to enforce us to make quite strong an assumption on the velocity, namely that u0 ∈ H 1 . We shall see later on how it can be removed. 4.1. Proof in the case β = 1. First step : Local well-posedness. Theorem 0.4 provides us with a local strong solution (a, u, ∇Π) which belongs to ETα,1 (see the statement of theorem 0.4). We denote by T ? ∈ (0, +∞] the maximum time of existence of (a, u, ∇Π), i.e the supremum of all T such that (a, u, ∇Π) belongs to ETα,1 . A standard continuation ? argument shows that (a, u, ∇Π) cannot belong to ETα,1 = +∞. ? unless T Second step : Existence of a global weak solution. In dimension N = 2, the e under the hypotheses that u0 ∈ H 1 existence of a global weak solution (e a, u e, ∇Π) ∞ and ρ0 ∈ L is bounded away from vacuum is well known (see e.g [1] or [16]). Actually besides the energy inequality (0.5) and the conservation of L∞ norm for + 1 2 + 1 e ∈ L2 (R+ ; L2 ) and the ρe±1 , we have ∇e u ∈ L∞ e, ∇Π loc (R ; H ) ∩ Lloc (R ; H ), ∂t u loc following inequality with some universal constant C : Z t Z t

√

e 2 2 2 2 −1 µ k∇e u(t)kL2 + k ρ∂t u ekL2 dτ + C ukL2 dτ

∇Π + µ2 k∆e 0

0

L2

2

≤ µ k∇u0 kL2 e

Ckρ0 k3 ∞ L µ2

Rt 0

ke uk4L4 dτ

.


13

Of course, the standard energy inequality (0.5) combined with Gagliardo-Nirenberg inequality yields bounds for the argument of the exponential. Therefore, we are given with the local smooth unique solution (a, u, ∇Π) and e which is global and satisfies the properties with (at least) another solution (e a, u e, ∇Π) described above. Third step : Loosing a priori estimates for transport equations. At this point, e Yet, only a part of there is no (known) way of proving uniqueness of (e a, u e, ∇Π). the assumptions on the data have been used. Indeed, we used that ρ0 ∈ L∞ while a0 = ρ0 − 1 is actually in H 1+α . Does anything persist of the initial smoothness ? To answer this question, we shall take advantage of the fact that e a solves a transport equation by the vector field u e which belongs to L2loc (R+ ; H 2 ). In view of proposition (3.1), the H 1+α regularity of e a would be preserved if ∇e u were also in L1loc (R+ ; L∞ ) but H 1 fails to be embedded in L∞ in dimension N = 2 ! Nevertheless, the following statement holds true : Proposition 4.1. Assume N = 2 and 0 ≤ s < 2.Let f0 ∈ H s and v ∈ L (0, T ; H 2 ). Then the transport equation ∂t f + v · ∇f = 0, f|t=0 = f0 , 1

has a unique solution f which belongs to C([0, T ]; H s− ) for all > 0. Moreover, kf (t)kH s− may be bounded in terms of , kf0 kH s and k∇vkL1 (H 1 ) . t

This proposition has been first proved by B. Desjardins in [10] and [12] under a slightly stronger assumption: v ∈ L2 (0, T ; H 2 ) and in the case of a bounded domain or in T2 . In [9], we do a systematic study of loosing a priori estimates for transport equations, and proposition 4.1 comes up as a particular case. Fourth step : Weak=strong uniqueness. According to the previous step, we have for all α0 < α, 0

+ 1 2 + 2 u e ∈ L∞ loc (R ; H ) ∩ Lloc (R ; H )

e a ∈ C(R+ ; H 1+α ),

e ∈ L2loc (R+ ; L2 ). and ∇Π

Slight modifications to the proof of uniqueness in section 3 enable us to get e ≡ (a, u, ∇Π) (e a, u e, ∇Π)

as long as (a, u, ∇Π) belongs to ETα,β , thus on the time interval [0, T ? [. Fifth step : Continuation. Assume that T ? < +∞. Applying proposition 3.1 with s = 2, p = 2, and using (0.5) yields kukLe∞ (H 1 ) + µbkukLe1 (H 3 ) + k∇ΠkLe1 (H 1 ) ≤ T

T

T

κ CAκT eCAT

RT 2 0 k∇u(t)kL2 dt µb

× ku0 kH 1 + µbAT kukLe1 (H 3− α2 ) T

with AT = 1 + b−1 kak e∞

LT (H

N +α 2 2

)

. Now, in view of proposition 4.1 and second

step, AT may be bounded by some number involving only the initial data (as T ? is assumed to be finite). On the other hand, the energy inequality (0.5) yields a uniform bound for ∇u in L2 (0, T ? ×R2 ). Therefore, for some constant C0 depending only on the initial data, we have (4.1) kukLe∞ (H 1 ) + µbkukLe1 (H 3 ) + k∇ΠkLe1 (H 1 ) ≤ C0 ku0 kH 1 + µbkukLe1 (H 3− α2 ) . T

T

T

T

¨ DANCHIN RAPHAEL

14

e 1 (H 3− α2 ) = [L e 1 (H 2 ), L e 1 (H 3 )] α , Young inequality yields As L T T T 2 kukLe1 (H 3− α2 ) T

2

≤ KkukLe1 (H 3 ) + K 1− α kukLe1 (H 2 ) , T T12 2 1− α ≤ KkukLe1 (H 3 ) + K T ? kukL2 ? (H 2 ) . T

T

Note that the last term is bounded in terms of the initial data and T ? . Therefore, plugging the above inequality in (4.1) with a small enough K shows that (a, u, ∇Π) stays in ETα,1 uniformly for all T < T ? . A standard continuation argument contradicts the hypothesis T ? < ∞. 4.2. General case β > 0. Let us now treat the case where β ∈ (0, 1). Once again, theorem 0.4 provides us with a strong unique solution (a, u, ∇Π) in ETα,β . In particular, a straightforward interpolation argument show that u belongs to L2 (0, T ; H 1+β ), thus to L2 (0, T ; H 1 ). This ensures that u(T0 ) belongs to H 1 for some T0 < T . Now, one may go along the proof in the case β = 1 and get a e (on the time interval [T0 , +∞[) with data unique global strong solution (e a, u e, ∇Π) e ≡ (a, u, ∇Π) on [T0 , T ]. Therefore (aT0 , uT0 ). Uniqueness ensures that (e a, u e, ∇Π) (a, u, ∇Π) may be continued globally. Remark 4.2. One can object that the above argument is rather inelegant. Indeed, it does not give much information on the possible growth of the solution. Yet it works ! References [1] S. Antontsev, A. Kazhikhov and V. Monakhov : Boundary value problems in mechanics of nonhomogeneous fluids. Translated from the Russian. Studies in Mathematics and its Applications, 22. North-Holland Publishing Co., Amsterdam, 1990. [2] J.-M. Bony : Calcul symbolique et propagation des singularit´ es pour les ´ equations aux d´ eriv´ ees partielles non lin´ eaires, Annales Scientifiques de l’´ ecole Normale Sup´ erieure, 14, pages 209-246 (1981). [3] J.-Y. Chemin : Th´ eor` emes d’unicit´ e pour le syst` eme de Navier-Stokes tridimensionnel, Journal d’Analyse Math´ ematique, 77, pages 25–50 (1999). [4] R. Danchin : Global existence in critical spaces for compressible Navier-Stokes equations, Inventiones Mathematicae, 141, pages 579–614 (2000). [5] R. Danchin : A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14, pages 953–988 (2001). [6] R. Danchin : Density-dependent incompressible fluids in critical spaces, submitted. [7] R. Danchin : Local and global well-posedness results for flows of inhomogeneous viscous fluids, preprint. [8] R. Danchin : Density-dependent incompressible fluids in critical spaces : the case of bounded domains, in progress. [9] R. Danchin : Remarks on transport and transport-diffusion equations, preprint. [10] B. Desjardins : Global existence results for the incompressible density-dependent NavierStokes equations in the whole space, Diff. and Int. Equations, 10, pages 587–598 (1997). [11] B. Desjardins : Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations, Diff. and Int. Equations, 10, pages 577–586 (1997). [12] B. Desjardins : Regularity results for two-dimensional flows of multiphase viscous fluids, Archiv for Rational Mechanics and Analysis, 137, pages 135–158 (1997). [13] H. Fujita and T. Kato : On the Navier-Stokes initial value problem I, Archive for Rational Mechanics and Analysis, 16, pages 269-315 (1964). [14] O. Ladyzhenskaya and V. Solonnikov : The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids, Journal of Soviet Mathematics, 9, pages 697–749 (1978).


15

[15] J. Leray : Sur le mouvement d’un liquide visqueux remplissant l’espace, Acta mathematica, 63, pages 193-248 (1934). [16] P.-L. Lions : Mathematical Topics in Fluid Dynamics, Vol. 1 Incompressible Models, Oxford University Press (1996). [17] T. Runst and W. Sickel : Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. de Gruyter Series in Nonlinear Analysis and Applications, 3. Walter de Gruyter & Co., Berlin (1996). ´ Paris 6, 175 rue du Chevaleret, 75013 Paris Laboratoire J.-L. Lions, Universite E-mail address: [email protected]