Existence of solutions for compressible fluid models of Korteweg type

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

S0294-1449(00)00056-1/FLA

EXISTENCE OF SOLUTIONS FOR COMPRESSIBLE FLUID MODELS OF KORTEWEG TYPE

Raphaël DANCHIN^a,Benoît DESJARDINS^b

aLaboratoire d’Analyse Numérique, Université Paris – VI, 4, place Jussieu, 75252 Paris cedex 05, France

bD.M.I., E.N.S., 45, rue d’Ulm, 75005 Paris, France Received 30 March 2000

ABSTRACT. – The purpose of this work is to prove existence and uniqueness results of suitably smooth solutions for an isothermal model of capillary compressible fluids derived by J.E. Dunn and J. Serrin (1985), which can be used as a phase transition model.

We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In a functional setting as close as possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence for solutions when the pressure law may present spinodal regions.

Uniqueness is also obtained.

Assuming a lower and upper control of the density, we also show the existence of weak solutions in dimension 2 near equilibrium. Finally, referring to the work of Z. Xin (1998) in the non-capillary case, we describe some blow-up properties of smooth solutions with finite total mass._2001 Éditions scientifiques et médicales Elsevier SAS

RÉSUMÉ. – On s’intéresse ici à des résultats d’existence et d’unicité de solutions pour un modèle de fluides compressibles isothermes avec capillarité. Ce modèle de transition de phase a été dérivé par J.E. Dunn et J. Serrin (1985).

Pour commencer, on montre que le problème de Cauchy est bien posé dans des espaces à régularité critique pour le scaling des équations. Pour des données initiales proches d’un état d’équilibre stable, on obtient l’existence globale (et l’unicité) de solutions dans un cadre fonctionnel aussi proche que possible de l’espace d’énergie physique. Pour des lois de pression plus générales (pouvant être décroissantes), on prouve des résultats locaux en temps.

En supposant que l’on dispose d’un minorant strictement positif et d’une borne supérieure pour la densité, on obtient l’existence de solutions faibles en dimension 2 pour des données initiales proches de l’équilibre. Enfin, en adaptant un travail de Z. Xin pour les fluides sans capillarité, on établit l’explosion de solutions régulières à masse totale finie.2001 Éditions scientifiques et médicales Elsevier SAS

E-mail addresses: danchin@ann.jussieu.fr (R. Danchin), desjardi@math.polytechnique.fr (B. Desjardins).

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1. Introduction

Let us consider a fluid of density ρ >0, velocity field u∈ R^d (d >2), entropy density s, energy density e, and temperature θ =(∂e/∂s)ρ. We are interested in the following model of compressible capillary fluid, which can be derived from a Cahn–

Hilliard like free energy (see the pioneering work by J.E. Dunn and J. Serrin in [11], and also [1,6,12])

∂tρ+div(ρu)=0, (1)

∂t(ρu)+div(ρu⊗u)=div(S+K), (2)

∂t

ρ

e+u²

2

+div

ρu

e+u² 2

=div(α∇θ )+div (S+K)·u, (3) where the viscous stress tensor S and the Korteweg stress tensor K read as

Si,j = λdivu−P (ρ , e)δi,j +2µD(u)i,j, (4) Ki,j =κ

2 1ρ²− |∇ρ|²δi,j−κ∂iρ∂jρ , (5) D(u)i,j =(∂iuj +∂jui)/2 being the strain tensor, and (λ, µ) the constant viscosity coefficients of the fluid. We require thatλandµsatisfyµ >0 andλ+2µ >0, which in particular covers the case whenλandµsatisfy Stokes’ lawdλ+2µ=0. The thermal conduction coefficient α is a given non negative function of the temperature θ and the surface tension coefficientκ >0 is assumed to be constant. In view of the first principle of thermodynamics, the entropy densityssolves

∂t(ρs)+div(ρsu)=1

θ div(α∇θ )+K:D(u)+2µD(u):D(u)+λ|div u|². (6) As a reasonable starting point of our analysis, we consider the scaled Van der Waals equation of state

P (ρ)=aρθ 8

3−ρ −3ρ θ

, (7)

wherea is a positive constant, and the critical density ρc and temperatureθc are equal to 1. Depending on the fixed temperatureθ, the pressure is a nondecreasing function of the density ρ or may present decreasing regions (spinodal regions) for some values of ρ, which are thermodynamically unstable. The above equation of state(7)ensures the presence of two basic states, a “liquid” one, and a “gaseous” one. Let us as in [20] put emphasis on the existence of steady solutions connecting a gas phase to a liquid phase through a smoothly varying density profile. When initial conditions involve densities in the unstable (spinodal) region, the two phases are expected to spontaneously separate.

For details on the derivation of the above Korteweg like model, we refer to [1,11,12,16].

In what follows, we do not consider thermal fluctuations so that the pressure p is a function ofρ only. The corresponding isothermal model which was also considered in [13,20] then reads as

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∂tρ+div(ρu)=0, (8)

∂t(ρu)+div(ρu⊗u)−µ1u−(λ+µ)∇div u+ ∇P (ρ)=div K+ρf, (9) where f is an exterior forcing term, supplemented with initial conditions

ρ_|t=0=ρ0>0 and ρu_|t=0=m0. (10) In a bounded domainΩ, we would have to precise the boundary conditions, namely homogeneous Dirichlet conditions for the velocity: u_|∂Ω=0 and Neumann conditions for the density: ∂nρ_|∂Ω=0. In order to simplify the presentation, we will focus on the whole space caseR^d(d>2)and study the well-posedness of (8) (9) for an initial density close enough to an equilibrium density ρ >¯ 0, or at least bounded away from vacuum, which is a major difficulty in most of compressible fluid models.

Before getting into the heart of mathematical results, we first derive the physical energy bounds of the above system in the case f≡0 to simplify the presentation. Let

¯

ρ >0 be a constant reference density, andπ defined by π(s)=s

Zs

¯ ρ

P (z)

z² dz−P (ρ)¯

¯ ρ

!

, (11)

so thatP (s)=sπ⁰(s)−π(s),π⁰(ρ)¯ =0, and

∂tπ(ρ)+div uπ(ρ)+P (ρ)div u=0 inD⁰ (0, T )×R^d. (12) Notice thatπ is convex as far as P is non decreasing (since P⁰(s)=sπ⁰⁰(s)), which is the case forγ-type pressure laws. Multiplying the equation of momentum conservation by u and integrating by parts overR^d, we obtain the following energy estimate

Z

R^d

1

2ρu²+ π(ρ)−π(ρ)¯ +κ 2|∇ρ|²

(t) dx

+ Zt 0

ds Z

R^d

µD(u):D(u)+(λ+µ)|div u|²dx 6

Z

R^d

|m0|² 2ρ0

+(π(ρ0)−π(ρ))¯ +κ 2|∇ρ0|²

dx. (13)

Indeed, in order to compute formally the contribution to energy of the capillary tensor K, we observe that

div K=κ ρ∇1ρ . (14)

In view of the above expression, this model can be understood as a diffuse interface model, in which surface tension takes place between level sets of the continuously varying density. As a matter of fact, the right hand side (14)can be rewritten up to a gradient term as the product between∇ρand1ρ, which, roughly speaking, respectively

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represent the normal direction and the curvature of the level sets of the density. As observed for instance in [1], formal analyses show that the sharp interface limit leads to the classical two-fluid problem. We obtain indeed

−^Z

R^d

u·div Kdx=^Z

R^d

κdiv(ρu) 1ρ dx=κ Z

R^d

∂t∇ρ· ∇ρ dx=κ d dt

Z

R^d

|∇ρ|² 2 dx.

It follows that assuming that the total energy is finite E0=^Z

R²

1

2ρ0u²₀+ π(ρ0)−π(ρ)¯ +κ 2|∇ρ0|²

<+∞, (15)

we have the a priori bounds

π(ρ)−π(ρ)¯ and ρ|u|²∈L^∞(0,∞;L¹(R^d)), (16)

∇ρ∈L^∞ 0,∞;L² R^d^d and ∇u∈L² (0,∞)×R^d^d². (17) Let us emphasize at this point that the above a priori bounds do not provide any L^∞ control on the density from below or from above. Indeed, even in dimension d =2, H¹(R^d)functions are not necessarily locally bounded. Thus, vacuum patches are likely to form in the fluid in spite of the presence of capillary forces, which are expected to smooth out the density.

2. Mathematical results

We wish to prove existence and uniqueness results of solutions to (8) (9) in functional spaces very close to energy spaces. In the caseκ =0 andp(ρ)=aρ^γ, witha >0 and γ >1, P.-L. Lions proved in [17,18] the global existence of weak solutions “à la Leray”

(ρ ,u)to(8) (9)forγ >3d/(d+2)and initial data(ρ0,m0)such that π(ρ0)−π(ρ)¯ and |m0|²

ρ0

∈L¹(R^d), (18) where we agree that m0=0 on {x∈R^d/ρ0(x)=0}. More precisely, he obtains the existence of global weak solutions(ρ ,u)to (8)–(10) such that

• ρ− ¯ρ ∈L^∞(0,∞;L²_γ(R^d)) (where L²_p(R^d) spaces are Orlicz spaces defined in [18]),

• u∈L²(0,∞; ˙H¹(R^d))^d (H˙^s being defined in Section 3), with in addition

• ρ ∈C([0,∞);L^p_loc(R^d))if 16p < γ,

• ρ|u|²∈L^∞(0,∞;L¹(R^d)),ρu∈C([0,∞);L^{2γ /(γ}_loc ⁺¹⁾(R^d)-weak),

• ρ ∈L^q_loc([0,∞)×R^d)forq=γ −1+2γ /d.

Moreover, the energy inequality(13)holds for almost everyt>0.

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Notice that the main difficulty for proving Lions’ theorem consists in strong compactness properties of the densityρinL^p_locspaces required to pass to the limit in the pressure termp(ρ)=aρ^γ. In the capillary caseκ >0, more a priori bounds are available for the density, which belongs toL^∞(0,∞; ˙H¹(R^d)). Hence, one can easily pass to the limit in the pressure term. However, in the remaining quadratic terms involving gradients of the density∇ρ⊗ ∇ρ(see(5)), we have been unable to pass to the limit.

Let us mention now that the existence of strong solutions is known since the works by H. Hattori and D. Li [13,14]. Notice that high order regularity in Sobolev spacesH^s is required, namely the initial data (ρ0,m0)are assumed to belong toH^s×H^s⁻¹with s >d/2+4. Moreover, they considered convex pressure profiles, which cannot cover the case of Van der Waals’ equation of state.

Here we want to investigate the well-posedness of the problem in critical spaces, that is, in spaces which are invariant by the scaling of Korteweg’s system. Recall that such an approach is now classical for incompressible Navier–Stokes equations (see, for example, [7] and the references therein) and yields local well-posedness (or global well-posedness for small data) in spaces with minimal regularity.

Let us explain precisely the scaling of Korteweg’s system. We can easily verify that, if(ρ ,u)solves (8) (9), so does(ρλ,uλ), where

ρλ(t, x)=ρ(λ²t, λx) and uλ(t, x)=λu(λ²t, λx), provided the pressure lawP has been changed intoλ²P.

DEFINITION 1. – We will say that a functional space is critical with respect to the scaling of the equation if the associated norm is invariant under the transformation (ρ ,u)7→(ρλ,uλ)(up to a constant independent ofλ).

This suggests us to choose initial data(ρ0,u0)in spaces whose norm is invariant by (ρ0,u0)7→(ρ0(λ·), λu0(λ·)).

A natural candidate is the homogeneous Sobolev spaceH˙^d/2×(H˙^d/2⁻¹)^d, but since H˙^d/2 is not included inL^∞, we cannot expect to get L^∞ control on the density when ρ0∈ ˙H^d/2. This is the reason why, instead of the classical homogeneous Sobolev spaces H˙^s(R^d), we will consider homogeneous Besov spaces with the same derivative index B^s=B_2,1^s (R^d)(for the corresponding definitions, we refer to Section 3). One of the nice property ofB^sspaces for critical exponentssis thatB^d/2is an algebra embedded inL^∞. This allows to control the density from below and from above, without requiring more regularity on derivatives ofρ.

Since a global in time approach does not seem to be accessible for general data, we will mainly consider the global well-posedness problem for initial data close enough to stable equilibria (Section 4). More precisely, we will state the following theorem:

THEOREM 1. – Letρ >¯ 0 be such that P⁰(ρ) >¯ 0. Suppose that the initial density fluctuation ρ0− ¯ρ belongs to B^d/2∩B^d/2⁻¹, that the initial velocity u0is in (B^d/2⁻¹)^d and that the exterior forcing term f is inL¹(R⁺;B^d/2⁻¹)^d. Then there exists a constant η >0 depending only onκ, µ, λ, ρ , P⁰(ρ)¯ andd, such that, if

kρ0− ¯ρkB^d/2⁻¹∩B^d/2+ ku0kB^d/2⁻¹+ kfkL¹(B^d/2⁻¹)6η,

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then (8)–(10) has a unique global solution (ρ ,u) such that the density fluctuation (ρ − ¯ρ) ∈ C(R⁺;B^d/2⁻¹∩B^d/2)∩L¹(R⁺;B^d/2⁺¹ ∩B^d/2⁺²) and the velocity u ∈ C(R⁺;B^d/2⁻¹)^d ∩L¹(R⁺;B^d/2⁺¹)^d.

In Section 5, we get a local in time existence result for initial densities bounded away from zero, which does not require any stability assumption on the pressure law, and thus applies to Van der Waals’ law. The precise statement reads as follows:

THEOREM 2. – Suppose that the forcing term f belongs to L¹_loc(R+; B^d/2⁻¹)^d, that the initial velocity u0 belongs to (B^d/2⁻¹)^d, and that the initial den- sity ρ0 satisfies (ρ0− ¯ρ) ∈B^d/2 and ρ0 >c for a positive constant c. Then there exists T >0 such that (8)–(10) has a unique solution (ρ ,u) satisfying (ρ − ¯ρ) ∈ C([0, T];B^d/2)∩L¹([0, T];B^d/2⁺²)and u∈C([0, T];B^d/2⁻¹)^d∩L¹([0, T];B^d/2⁺¹)^d. In Section 6, we show that the problem is still locally well-posed in more general scaling invariant Besov spaces of type B_p,1^s which are not related to energy spaces (namely ρ − ¯ρ is assumed to be in B_p,1^d/p and u0 to be in (B_p,1^d/p⁻¹)^d). No stability assumption on the pressure is required, but we have to suppose that the density is close to a constant (see Theorem 5). Let us observe that working withp > d allows to consider initial velocities inB_p,1^s spaces with negative exponentss, which is in particular relevant for oscillating initial data.

Finally, we will investigate blow-up properties of smooth solutions without smallness assumptions on the data, like in the work of Z. Xin [22], and study sufficient conditions for the existence of weak solutions close to equilibria in dimensiond=2.

Notation. In all the paper, C will stand for a “harmless” constant, and we will sometimes use the notationA.Bequivalently toA6CB.

3. Littlewood–Paley theory and Besov spaces 3.1. Littlewood–Paley decomposition

The homogeneous Littlewood–Paley decomposition relies upon a dyadic partition of unity. We can use for instance anyϕ∈C^∞(R^d), supported inC^def={ξ ∈R^d,3/46|ξ|6 8/3}such that

X

`∈Z

ϕ 2⁻^`ξ=1 ifξ 6=0.

Denotingh=F⁻¹ϕ, we then define the dyadic blocks by 1`u^def=ϕ 2⁻^`Du=2^`d

Z

R^d

h 2^`yu(x−y) dy and S`u= ^X

k6`−1

1ku.

The formal decomposition

u=^X

`∈Z

1`u (19)

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is called homogeneous Littlewood–Paley decomposition. Let us observe that the above formal equality does not hold inS⁰(R^d)for two reasons:

(i) The right-hand side does not necessarily converge inS⁰(R^d).

(ii) Even if it does, the equality is not always true inS⁰(R^d)(consider the caseu=1).

Nevertheless, (19) holds true modulo polynomials (see [21]).

Furthermore, the above dyadic decomposition has nice properties of quasi-orthogonal- ity: with our choice ofϕ, we have

1k1`u≡0 if|k−`|>2, and 1k(S`−1u1`u)≡0 if|k−`|>5. (20) 3.2. Homogeneous Besov spaces

DEFINITION 2. – Fors∈R,p∈ [1,+∞],q∈ [1,+∞]andu∈S⁰(R^d), we set kukB_p,q^s

def=^X

`∈Z

2^s`k1`ukL^p

q1/q

.

A difficulty due to the choice of homogeneous spaces arises at this point. Indeed, k·kB^s_p,q cannot be a norm on {u ∈ S⁰(R^d),kukB_p,q^s < +∞} because kukB_p,q^s = 0 means that u is a polynomial. This enforces us to adopt the following definition for homogeneous Besov spaces (see [5] for more details):

DEFINITION 3. – Lets∈R,p∈ [1,+∞]andq∈ [1,+∞]. Denotem= [s−d/p]if s−d/p /∈Zorq >1 andm=s−d/p−1 otherwise. Ifm <0, then we defineB_p,q^s as

B_p,q^s =

u∈S⁰ R^d| kukB_p,q^s <∞andu=^X

`∈Z

1`uinS⁰ R^d.

Ifm>0, we denote byPm[R^d]the set of polynomials of degree less than or equal tom and we set

B_p,q^s =

u∈S⁰ R^d/Pm

R^d| kukB^s_p,q<∞ and u=^X

`∈Z

1`uinS⁰ R^dPm

R^d.

Remark 1. – The above definition is a natural generalization of the homogeneous Sobolev or Hölder spaces: one can show that B_∞^s _,_∞ is the homogeneous Hölder space C˙^s and thatB_2,2^s is the homogeneous Sobolev spaceH˙^s.

In the sequel, we will use only Besov spacesB_p,q^s withq=1 and we will denote them byB_p^s or even byB^s if there is no ambiguity on the indexp.

3.3. Basic properties of Besov spaces

PROPOSITION 1. – The following properties hold:

(i) Density: ifp <+∞and|s|6d/p, thenC₀^∞is dense inB_p^s. (ii) Derivation: there exists a universal constantC such that

C⁻¹kukB^s_p6k∇uk_B_p^s−16CkukB_p^s.

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(ii⁰) Fractional derivation: let Λ^def=√

−1 and σ ∈R. Then the operator Λ^σ is an isomorphism fromB_p^s toB_p^s⁻^σ.

(iii) Sobolev embeddings: ifp1< p2 thenB_p^s

1,→B_p^s⁻^d(1/p¹⁻^1/p²⁾

2 (where,→means

continuous embedding).

(iv) Algebraic properties: fors >0,B_p^s ∩L^∞is an algebra.

(v) Interpolation: (B_p^s¹, B_p^s²)θ,1=B_p^{θ s}¹⁺⁽¹⁻^{θ )s}².

In Section 6, we will make extensive use of the spaceB_p^d/p. Note that, ifp <+∞, thenB_p^d/p is an algebra included in the spaceC0of continuous functions which tend to 0 at infinity. Note also that B_p^d/p×(B_p^d/p⁻¹)^d is invariant by the scaling of Korteweg’s system.

In Sections 4 and 5, we will focus on the casep=2. Note that the following inclusion chain

B_2,1^d/2,→ ˙H^d/2=B_2,2^d/2,→B_2,^d/2_∞

shows us thatH˙^d/2is very close toB_2,1^d/2. ButB_2,1^d/2has two additional nice properties: to be an algebra and to be a subset ofC0.

3.4. Besov–Chemin–Lerner spaces

The study of non stationary PDE’s usually requires spaces of typeL^r_T(X)^def=L^r(0, T; X)for appropriate Banach spacesX. In our case, we expectX to be a Besov space, so that it is natural to localize the equations through Littlewood–Paley decomposition. We then get estimates for each dyadic block and perform integration in time. But, in doing so, we obtain bounds in spaces which are not of type L^r(0, T;B_p^s). This approach was initiated in [9] and naturally leads to the following definitions:

DEFINITION 4. – Let(ρ , p)∈ [1,+∞]²,T ∈ ]0,+∞]ands∈R. We set kuk_L_e^ρ

T(B^s_p) def=^X

`∈Z

2^`s Z^T

0

k1`u(t)k^ρL^pdt 1/ρ

. Noticing that Minkowski’s inequality yields kukL^ρ_T(B_p^s) 6 kuk_L_e^ρ

T(B^s_p), we define Le^ρ_T(B_p^s)spaces as follows

Le^ρ_T B_p^s^def=u∈L^ρ_T B_p^s| kuk_e_L^ρ

T(B_p^s)<+∞ .

Let us observe that L¹_T(B_p^s)=L^e¹_T(B_p^s) but that the embedding L^e^ρ_T(B_p^s)⊂L^ρ_T(B_p^s) is strict ifρ >1.

We will denote byC^eT(B_p^s) the subset of functions ofL^e^∞_T (B_p^s)which are continuous on[0, T]with values inB_p^s.

Throughout the paper, the notation Lê^ρ_T(B_p^s ∩ B_p^s⁰0) (respectively Le^ρ_T(B_p^s×B_p^s⁰0)) will stand forLê^ρ_T(B_p^s)∩Lê^ρ_T(B_p^s⁰0)(respectivelyLê^ρ_T(B_p^s)×Lê^ρ_T(B_p^s⁰0)). More- over, in the caseT = +∞, theT will be omitted. For example,Lê^ρ(B_p^s)meansLê^ρ_+∞(B_p^s).

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We will often use the following interpolation property kuk_e_L^ρ

T(B_p^s)6kuk^θ_e_L^ρ1

T (B^s_p¹)kuk¹_e⁻^θ

L^ρ_T²(B^sp²) with 1 ρ = θ

ρ1

+1−θ ρ2

ands=θ s1+(1−θ )s2, and the following embeddings

Le^ρ_T B_p^d/p,→L^ρ_T(C0) and C^eT B_p^d/p,→C [0, T] ×R^d.

TheL^e^ρ_T(B_p^s)spaces suit particularly well to the study of smoothing properties of the heat equation. In [7], J.-Y. Chemin proved the following proposition

PROPOSITION 2. – Letp∈ [1,+∞]and 16ρ26ρ16+∞. Letusolve ∂tu−ν1u=f,

u_|t=0=u0.

Then there existsC >0 depending only ond,ν,ρ1andρ2such that kuk_e_L^ρ1

T (B^s+2/ρ_p ¹)6Cku0kB_p^s +Ckfk_e_L^ρ2

T(B_p^s−2+2/ρ²).

In Sections 4, 5 and 6, we will point out similar smoothing properties for the linearized Korteweg system.

Let us now state properties ofL^e^ρ_T(B_p^s)spaces with respect to the product.

PROPOSITION 3. – Ifs >0, 1/ρ2+1/ρ3=1/ρ1+1/ρ4=1/ρ61,u∈L^ρ_T¹(L^∞)∩ Le^ρ_T³(B_p^s)andv∈L^ρ_T²(L^∞)∩L^e^ρ_T⁴(B_p^s), thenuv∈L^e^ρ_T(B_p^s)and

kuvk_e_L^ρ

T(B_p^s).kukL^ρ_T¹(L^∞)kvk_e_L^ρ⁴

T (B_p^s)+ kvkL^ρ_T²(L^∞)kuk_L_e^ρ³

T (B_p^s).

If s1, s26d/p, s1+s2>0, 1/ρ1+1/ρ2=1/ρ61, u∈Lê^ρ_T¹(B_p^s¹) and v∈Lê^ρ_T²(B_p^s²), thenuv∈Lê^ρ_T(B_p^s¹⁺^s²⁻^d/p)and

kuvk_e_L^ρ

T(B_p^s¹^+s^2−d/p).kuk_L^ρ¹

T (B_p^s¹)kvk_L^ρ²

T (B^s_p²).

This proposition is a straightforward adaptation of the corresponding results for usual homogeneous Besov spaces (see [8]).

We finally need a composition lemma inL^e^ρ_T(B_p^s)spaces.

LEMMA 1. – Lets >0,p∈ [1,+∞]andu∈L^e^ρ_T(B_p^s)∩L^∞_T (L^∞).

(i) LetF ∈W_loc^[^s^]+^2,^∞(R^d)such thatF (0)=0. ThenF (u)∈L^e^ρ_T(B_p^s). More precisely, there exists a functionCdepending only ons,p,d andF such that

kF (u)k_e_L^ρ

T(B_p^s)6C kukL^∞_T(L∞)

kuk_L_e^ρ

T(B^s_p).

(ii) Ifvalso belongs toL^e^ρ_T(B_p^s)∩L^∞_T (L^∞)andG∈W_loc^[^s^]+^3,^∞(R^d), thenG(v)−G(u) belongs toL^e^ρ_T(B_p^s)and there exists a function Cdepending only ons,p,d andG, and such that

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G(v)−G(u)_e_L^ρ

T(B^s_p)

6C kukL^∞_T(L^∞),kvkL^∞_T(L^∞)

kv−uk_e_L^ρ

T(Bp^s)(1+ kukL^∞_T(L^∞)+ kvkL^∞_T(L^∞)) + kv−ukL^∞_T(L^∞)(kuk_e_L^ρ

T(B_p^s)+ kvk_e_L^ρ

T(B_p^s)).

Proof. – For (i), one just has to use the proof of [2] and replace L² norms withL^p norms. For (ii), we use the following identity

G(v)−G(u)=(v−u) Z1 0

H u+τ (v−u)dτ+G⁰(0)(v−u),

whereH (w)=G⁰(w)−G⁰(0), and we conclude by using (i) and Proposition 3. 2 4. Global solutions near equilibrium

In this section, we want to prove global existence and uniqueness of suitably smooth solutions to the Korteweg system(8) (9)in the functional spacesL^e^ρ_T(B₂^s)which are very close to the physical energy spaces. Given a reference density ρ¯ such that the stability condition P⁰(ρ) >¯ 0 is satisfied, we introduce the density fluctuation q =(ρ− ¯ρ)/ρ¯ and the scaled momentum m=ρu/ρ. We also define the scaled viscosity coefficients¯

¯

µ=µ/ρ¯ andλ¯ =λ/ρ¯and the scaled surface tension coefficientκ¯ = ¯ρκ. Assuming that the density ρ is bounded away from zero, we rewrite the Korteweg system (8) (9) as follows

∂tq+div m=0, (21)

∂tm− ¯µ1m−(¯λ+ ¯µ)∇div m− ¯κ∇1q+P⁰(ρ)¯ ∇q=G(q,m)+f, (22)

(q,m)_|t=0=(q0,m0), (23)

where we define G=G1+G2+G3+G4+G5 by G1(q,m)= −div

m⊗m 1+q

, G2(q,m)= −∇H (q), G3(q,m)= − ¯µ1

qm 1+q

−(¯λ+ ¯µ)∇div qm

1+q

, G4(q,m)=κ¯

2∇ 1q²− |∇q|²− ¯κdiv(∇q⊗ ∇q)= ¯κq∇1q, and

G5(q,m)=fq,

H being defined byH (q)=(P (ρ(1¯ +q))−P (ρ)¯ −P⁰(ρ)q¯ ρ)/¯ ρ.¯

In Section 4.1, we study the linearized system around(q,m)=(0,0), which turns out to have the same smoothing properties as the heat equation. Finally, we prove in Section 4.2 our main global theorem, estimating the right-hand side G of(22)in terms of suitable norms of(q,m). Notice that in Sections 4 and 5,B^s will stand forB_2,1^s .

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4.1. Estimates for the linearized system

This section is devoted to the linearized isothermal system of Korteweg type around (q,m)=(0,0). This system reads

∂tq+div m=F,

∂tm− ¯µ1m−(λ¯+ ¯µ)∇div m− ¯κ∇1q+β∇q=G. (LNSK1) The term β∇q corresponds to the linearized pressure (that is β =P⁰(ρ) >¯ 0). Our purpose is to prove estimates for (LNSK1) in Besov spaces closely related to energy spaces. We get:

PROPOSITION 4. – Let s ∈R, 16r16r 6+∞ and T ∈ ]0,+∞]. If (q0,m0)∈ (B^s∩B^s⁻¹)×(B^s⁻¹)^d and(F,G)∈L^e^r_T¹((B^s⁻²⁺^2/r¹∩B^s⁻³⁺^2/r¹)×(B^s⁻³⁺^2/r¹)^d)then the linear system (LNSK1) has a unique solution(q,m)∈C^eT((B^s∩B^s⁻¹)×(B^s⁻¹)^d)∩ Le^r_T((B^s⁺^2/r∩B^s⁻¹⁺^2/r)×(B^s⁻¹⁺^2/r)^d). Moreover, there exists a constantC depending only onr,r1,µ,¯ λ,¯ κ¯ andβ such that the following inequality holds:

kqk_e_L^r

T(B^s⁺^2/r∩B^s⁻¹⁺^2/r)+ kmk_e_L^r

T(B^s⁻¹⁺^2/r)

6C kq0kB^s∩B^s⁻¹+ km0kB^s⁻¹+ kFk_e_L^r¹

T(B^s⁻²⁺^2/r1∩B^s⁻³⁺^2/r1)+ kGk_e_L^r¹

T(B^s⁻³⁺^2/r1)

. Proof. – Denote by W (t) the semi-group associated to (LNSK1). According to Duhamel’s formula,

q(t) m(t)

=W (t) q0

m0

+

Zt 0

W (t−s) F (s)

G(s)

ds. (24)

Let us first consider the case F ≡ 0 and G ≡ 0 and denote (q`(t),m`(t))^t = W (t)(1`q0, 1`m0)^t. Then, we have the following lemma

LEMMA 2. – There exist two positive constantscand C depending only onλ,¯ µ,¯ κ¯ andβ such that for all`∈Z,

km`(t)kL²+ k∇q`(t)kL² + kq`(t)kL²

6Ce⁻^c2^2`^t k1`m0kL²+ k∇1`q0kL²+ k1`q0kL²

.

Proof. – We apply the operator1`to (LNSK1) in the caseF ≡G≡0 and get

∂tq`+div m`=0, (25)

∂tm`− ¯µ1m`−(¯λ+ ¯µ)∇div m`− ¯κ∇1q`−β∇q`=0. (26) In view of Eq. (25), integrations by parts yield

Z

R^d

1q`div m`dx=1 2

d

dtk∇q`k²_L² and Z

R^d

q`div m`dx= −1 2

d

dtkq`k²_L².

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Thus, taking scalar product of (26) with m`, we get 1

2 d

dt km`k²_L²+βkq`k²_L²+ ¯κk∇q`k²_L²+ ¯µk∇m`k²_L²+(λ¯+ ¯µ)kdiv m`k²_L² =0. (27) In order to obtain a second energy estimate, we take the scalar product of m`with the gradient of (25), which yields

Z

R^d

m`·∂t∇q`dx− kdiv m`k²_L² =0. (28) Taking the scalar product of (26) with∇q`, we obtain

Z

R^d

∇q`·∂tm`dx+ ¯κk1q`k²L² +βk∇q`k²L²6Ck∇m`kL²k∇²q`kL². (29)

Summing (28) and (29), we deduce d

dt Z

R^d

m`· ∇q`dx

+κ¯

2k∇²q`k²_L²+βk∇q`k²_L²6Ck∇m`k²_L². (30) Letα >0 be a constant to be chosen later and denote

h²_`= km`k²L²+ ¯κk∇q`k²L²+βkq`k²L²+2α Z

R^d

m`· ∇q`dx.

As a result, from (30) and (27), we derive for some positive constantc0

1 2

d

dth²_`+c0 k∇m`k²_L²+α∇²q`²

L²+αk∇q`k²_L²

6Cαk∇m`k²L². (31) Now choosingαsuitably small, we deduce that

1

δh²_`6km`k²_L2+ ¯κk∇q`k²_L2+βkq`k²_L2 6δh²_`, (32) for some positiveδ. Thus, there exists a constant c >0 such that

1 2

d

dth²_`+c2^2`h²_`60, so that the proof of Lemma 2 is complete. 2

Proof of Proposition 4 (continued). – In view of Lemma 2 and formula (24), we have k1`m(t)kL² + k∇1`q(t)kL²+ k1`q(t)kL²

6Ce⁻^c2^2`^t k1`m0kL² + k∇1`q0kL²+ k1`q0kL²