Local strong solution for the incompressible Korteweg model

Local strong solution for the incompressible Korteweg model. � Solution local fort pour le mod`ele de Korteweg incompressible. D. Bresch aF. Guill´en-Gonz´alez bJ. Lemoine cM. A. Rodr´ıguez-Bellido bM. Sy d aLaboratoire de Mod´elisation et Calcul, Universit´e Joseph-Fourier, 51 Rue des Math´ematiques, 38051 Grenoble, France bDpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apto. 1160, 41080 Sevilla, Spain cLaboratoire de Math´ematiques & CNRS 6620. Univ. Blaise Pascal, 63177 Aubi`ere, France dLaboratoire d’Analyse Num´erique et d’Informatique. Univ. Gaston Berger, Saint-Louis, S´en´egal Abstract We consider a Navier-Stokes type system with a Korteweg stress tensor, coupled with a concentration equation without diffusion. We give a result concerning the local in time existence (and uniqueness) of strong solution for any data (and the global in time existence defined in a interval (0,T) for T<+∞fixed if data are small enough), in the case of a bounded domain Ω⊂R3.To cite this article: D. Bresch & al. , C. R. Acad. Sci. Paris, Ser. I ???. R´esum´e On consid`ere un syst`eme de type Navier-Stokes avec un tenseur de Korteweg, coupl´e avec une ´equation de concentration sans diffusion. On donne un premier r´esultat d’existence local en temps (et unicit´e) de solution forte pour des donn´ees quelconques (et d’existence globale en temps defini dans un interval (0,T) pour T<+∞ fix´e si les donn´ees sont petites), dans le cas d’un domaine born´e Ω⊂R3.Pour citer cet article : D. Bresch & al., C. R. Acad. Sci. Paris, Ser. I ? ? ?. Version fran¸caise abr´eg´ee On s’int´eresse `a l’´etude de la r´egularit´e de solution d’un mod`ele de type Navier-Stokes avec un tenseur des contraintes sp´ecifiques, introduit par Korteweg en 1901. Un tel mod`ele propose que la distribution de densit´e non uniforme (de concentration ou de temp´erature) induit des efforts et de la convection dans �The second and fourth authors have been partially supported by D.G.E.S. and M.C. y T. (Spain), Projet BFM2003- 06446-C02-01. Email addresses: [email protected] (D. Bresch), [email protected] (F. Guill´en-Gonz´alez), [email protected] (J. Lemoine), [email protected] (M. A. Rodr´ıguez-Bellido), [email protected] (M. Sy). Preprint submitted to Elsevier Science 14 octobre 2005 le fluide. Dans [4], il ´etudient le cas de deux liquides miscibles, dans lequel le tenseur est une tension interfaciale effective qui se relaxe avec le temps dˆu `a l’effet de la diffusion. Ici, on consid`ere un mod`ele similaire mais sans diffusion. Ainsi, il est plus difficile d’obtenir la r´egularit´e forte du mod`ele. Concr`etement, le mod`ele qu’on ´etudie est dans un domaine Ω⊂R3,suffisamment r´egulier, dont les inconnues sont la vitesse u(t, x), la pression π(t, x), et la concentration ρ(t, x). Le mod`ele est donn´e par (KM) (voir apr`es). On prouve le r´esultat suivant : Theorem 0.1 Si ρ0∈W2,p(Ω)et u0∈W1,p(Ω), il existe T∗∈(0,T]suffisamment petit ou T∗=Tet des donn´ees petites, telle qu’il existe une (seule) solution de (KM)dans (0,T ∗)avec ρ∈L∞(0,T ∗;W2,p(Ω)), u∈Lp(0,T ∗;W2,p(Ω)) ∩L∞(0,T ∗;W1,p(Ω)) and ∂tu∈Lp(0,T ∗;Lp(Ω)). Les techniques pour l’obtention de solution r´eguli`ere du mod`ele de type Navier-Stokes `a densit´e variable ne peuvent pas s’appliquer dans ce cas car le couplage est fortement non lin´eaire, dˆu au terme ∆ρ∇ρ, dans le syst`eme pour u. Pour le traiter d’une fa¸con convenable, on a utilis´e des estimations de type non-hilbertien [5,7] et des techniques de point fixe. 1. Introduction In this work, we are interested in the study of regularity for a Navier-Stokes type model with a specific stress tensor, introduced for the first time by Korteweg, [3]. He proposed in 1901 that a nonuniform density (concentration or temperature) distribution could induce stresses and convection in a fluid. In [4], the authors study the case of two miscible liquids brought in contact, this tensor acts like an effective interfacial tension that relaxes with time due to the mass diffusion. We take a similar model that does not consider diffusion for the concentration ρ. Thus, it becomes more difficult to obtain the strong regularity. Such work is the aim of this paper. The unknowns are the solenoidal velocity u(t, x), the pressure π(t, x), and the concentration ρ(t, x). Moreover, we suppose that the fluid is confined in a three-dimensional smooth enough domain Ω. Denoting Q=(0,T)×Ω,Σ=(0,T)×∂Ω, the model can be written as: (KM)         ∂tu−ν∆u+u·∇u+∇π=−k∇ρ∆ρin Q, ∂tρ+u·∇ρ=0,∇·u=0 inQ, u=0on Σ,u|t=0 =u0,ρ|t=0 =ρ0in Ω. As for the coefficients, ν>0 is the fluid viscosity, λ>0 is the elasticity constant and γ>0 is a relaxation in time constant. On the other hand, ρ0and u0are the initial data. Note that the coupling term depending on ρthat appears in the system for ucan be rewritten as kρ∇(∆ρ), using the following manipulation: ρ∇∆ρ=∇(ρ∆ρ)−∇·(∇ρ⊗∇ρ)+1 2∇(|∇ρ|2)=∇(ρ∆ρ)−∇ρ∆ρ, and including the term ∇(ρ∆ρ) in the pressure term. Moreover, a conservative formulation of this term is −k∇·(∇ρ⊗∇ρ), including ∇�1 2|∇ρ|2�in the pressure term. 1.1. Functional spaces and equivalent norms We denote by Hand Vthe space of type L2and H1respectively, associated to the incompressibility and Dirichlet homogeneous boundary conditions for the velocity: H={u∈L2(Ω):∇·u=0,u·n|∂Ω=0}V={u∈H1(Ω):∇·u=0,u|∂Ω=0}. 2 Moreover, we will do the estimates in W1,p(Ω)∩Vor W2,p(Ω)∩V, and we will take �∇v�Lp(Ω)and �∆v�Lp(Ω)as the equivalent norms in those spaces. In order to study the concentration of the fluid ρ,we consider the space: Wk,p N(Ω)={ρ∈Wk,p(Ω)/� Ω ρ(t)dΩ=� Ω ρ0dΩ,∀t}≡ρ0+Wk,p N,0(Ω)(k=1,2,p∈[1,∞)) where Wk,p N,0(Ω)={ρ∈Wk,p(Ω)/�Ωρ(t)dΩ=0,∀t}.Thus,�∇ρ�Lp(Ω)and �∆ρ�Lp(Ω)define norms in W1,p N(Ω) and W2,p N(Ω), respectively. Observe that the bold face notation is used for the vectorial spaces in order to make different from the scalar ones. 1.2. Previous results Notice that under (weak) regularity hypothesis for the data u0∈Hand ρ0∈H1(Ω), this model has global in time weak estimates, i. e. any weak solution (if exists) verifies has the following estimates: u∈L∞(0,T;H)∩L2(0,T;V),ρ∈L∞(0,T;H1(Ω)).(1) Observe that those weak estimates are obtained taking the gradient of the equation of ρand multiplying by k∇ρ; taking uas a test function for the equation of u, and adding both expressions. That reads: 1 2 d dt �k�∇ρ�2 L2(Ω)+�u�2 L2(Ω)�+ν�∇u�2 L2(Ω)=0. However, these estimates are not regular enough in order to provide the convergence (in any sense) for the nonlinear term ∇ρ∆ρ(even for their conservative formulation ∇·(∇ρ⊗∇ρ)). 2. Uniqueness criterium Let (ρ1,u1,π1) and (ρ2,u2,π2) be two solutions of (KM). If we call ¯ρ=ρ1−ρ2and ¯ u=u1−u2,we can write the system verified by (¯ρ,¯ u,¯π) as follows: (KMu)         ∂t¯ρ+(u1·∇)¯ρ+( ¯ u·∇)ρ2=0 inQ, ∂t¯ u−ν∆¯ u+(u2·∇)¯ u+( ¯ u·∇)u1+∇¯π+k∆ρ1∇¯ρ+k∆¯ρ∇ρ2=0in Q, ∇·¯ u=0inQ,¯ u=0on Σ,¯ u|t=0 =0,¯ρ|t=0 =0 inΩ, Thus, taking gradient in (KMu)1,multiplyingby−k∇¯ρand multiplying (KMu)2by ¯ u, we obtain: 1 2 d dt�k�∇¯ρ�2 L2(Ω)+�¯ u�2 L2(Ω)�+ν�∇¯ u�2 L2(Ω)=−� Ω (¯ u·∇)u1·¯ udΩ−k� Ω (¯ u·∇)¯ρ∆ρ1dΩ The right hand side terms can be bounded as follows: � Ω (¯ u·∇)u1·¯ udΩ≤� ¯ u�2 L4(Ω)�∇u1�L2(Ω)≤¯ε�∇¯ u�2 L2(Ω)+C¯ε�∇u1�4 L2(Ω)�¯ u�2 L2(Ω) k� Ω (¯ u·∇)¯ρ∆ρ1dΩ≤k�¯ u�L6(Ω)�∇¯ρ�L2(Ω)�∆ρ1�L3(Ω)≤¯ε�∇¯ u�2 L2(Ω)+C¯ε�∆ρ1�2 L3(Ω)�∇¯ρ�2 L2(Ω) Therefore, if we want to conclude the uniqueness of weak solution by means of a Gronwall’s Lemma, we need to impose that ∇u1∈L4(0,T;L2(Ω)) and ρ1∈L2(0,T;W2,3(Ω)). 3 3. Stronger estimates We want to obtain strong estimates for u,πand ρ. Using the non-hilbertian estimates for the Stokes problem [7], we found that for any p>1: �∂tu�p Lp t(Q)+�u�p Lp t(W2,p(Ω)) +�π�p Lp t(W1,p(Ω)) ≤C��(u·∇)u�p Lp t(Q)+�∆ρ∇ρ�p Lp t(Q)�(2) We bound the last term of (2) as �∆ρ∇ρ�p Lp(Ω)≤ �∇ρ�p L∞(Ω)�∆ρ�p Lp(Ω)≤C�∆ρ�2p Lp(Ω), for any p>3. In order to bound �∆ρ�p Lp(Ω), we take the laplacian of (KM)2and multiply by |∆ρ|p−2∆ρ, obtaining: 1 p d dt�∆ρ�p Lp(Ω)≤� Ω |∆u||∇ρ||∆ρ|p−1dΩ+� Ω |∇u||D2ρ||∆ρ|p−1dΩ +� Ω (u·∇)∆ρ|∆ρ|p−2∆ρdΩ:= I1+I2+I3(= 0) = I1+I2 We bound the Ii-terms as follows (recall that p>3): I1≤�∆u�Lp(Ω)�∇ρ�L∞(Ω)�∆ρ�p−1 Lp(Ω)≤C�∆u�Lp(Ω)�∆ρ�p Lp(Ω) I2≤C�∇u�L∞(Ω)�D2ρ�p Lp(Ω)≤C�u�W2,p(Ω)�∆ρ�p Lp(Ω) which led us to the expression d dt �∆ρ(t)�p Lp(Ω)≤C�u�W2,p(Ω)�∆ρ(t)�p Lp(Ω). Thus, integrating directly we obtain: �∆ρ�p L∞ t(Lp(Ω)) ≤�∆ρ0�p Lp(Ω)eC�u�L1 t(W2,p),∀t>0.(3) Therefore, integrating in time the estimates for �∇ρ∆ρ�p Lp(Ω)and using (3), we obtain: t � 0 �∆ρ∇ρ�p Lp(Ω)(s)ds ≤Ct�∆ρ�2p L∞ tLp(Ω)≤Ct�ρ0�2p W2,p(Ω)e2C�u�L1 t(W2,p(Ω)) On the other hand, the nonlinear term in ucan be estimated as: t � 0 �(u·∇)u�p Lp(Ω)(s)ds ≤ t � 0 �u(s)�p L∞(Ω)�∇u(s)�p Lp(Ω)ds ≤C t � 0 �u(s)�2p W1,p(Ω)ds ≤Ct�u�2p L∞ t(W1,p(Ω)) Returning to inequality (2), it becomes the following inequality: �∂tu�p Lp t(Lp(Ω)) +�u�p Lp t(W2,p(Ω)) +�π�p Lp t(W1,p(Ω)) ≤Ct�u�2p L∞ t(W1,p(Ω)) +Ct�ρ0�2p W2,p(Ω)e2C�u�L1 t(W2,p(Ω)) (4) Observe that it is possible to show that (see Appendix): �u�p L∞ t(W1,p(Ω)) ≤�u0�p W1,p(Ω)+C��∂tu�p Lp t(Lp(Ω)) +�u�p Lp t(W2,p(Ω))�.(5) Thus, the inequality (4) can be rewritten in the following form: �u�p L∞ t(W1,p(Ω)) +�∂tu�p Lp t(Lp(Ω)) +�u�p Lp t(W2,p(Ω)) ≤�u0�p W1,p(Ω) +Ct��u�2p L∞ t(W1,p(Ω)) +�ρ0�2p W2,p(Ω)exp �2C�u�L1 t(W2,p(Ω))�� (6) 4 4. The main result Theorem 4.1 If ρ0∈W2,p(Ω)and u0∈W1,p(Ω), there exist a time T∗∈(0,T]either small enough or T∗=Tand small enough data, such that there is a (unique) solution of (KM)in (0,T ∗)with ρ∈ L∞(0,T ∗;W2,p(Ω)),u∈Lp(0,T ∗;W2,p(Ω)) ∩L∞(0,T ∗;W1,p(Ω)) and ∂tu∈Lp(0,T ∗;Lp(Ω)). Proof: We define the Banach space X(T)={u/u∈Lp(0,T;W2,p(Ω)) ∩L∞(0,T;W1,p(Ω)),∂tu∈ Lp(0,T;Lp(Ω))}endowed with the norm �u�X(T)=��u�p Lp T(W2,p)+�u�p L∞ T(W1,p)+�∂tu�p Lp T(Lp)�1/p . The argument we use is the following: Given a closed set Ar,Ar={u:�u�p X(T)≤r}⊂X(T) (r>0 will be chosen), where we define an operator R:A⊂E→Eas ¯ u∈A�→ R(¯ u)=ubeing E=L2(0,T;H1(Ω)) and uthe solution of the following linear problem (of Stokes type): ∂tu−ν∆u+∇p=−(¯ u·∇)¯ u−k∇ρ∆ρ,∇·u=0in Q,u|Σ=0,u|t=0 =u0in Ω, (7) where the concentration ρappearing on the right-hand side of (7)1is the solution of the transport problem: ∂tρ+( ¯ u·∇)ρ=0inQ,ρ|t=0 =ρ0in Ω. (8) Then, a regular (and unique) solution of the problem is a fixed point of operator R.Itsuffices, in order to apply the Schauder’s fixed point theorem, to verify the following statements: 1) Aris a convex compact set of E (for each r>0): Observe that Aris a convex closed subspace of X(T). The compact results of Aubin-Lions type ([6]) let us to insert Arcompactly in E. 2) There exists r>0such that R(Ar)⊂Ar:Suppose that ¯ u∈Ar. The main idea is to find r>0 such that the associated convex set Arverifies R(Ar)⊂Ar. Using (6), we get: �u�p X(T)≤�u0�p W1,p(Ω)+CT�r2+�ρ0�2p W2,p(Ω)exp �2CTp−1 pr�� Therefore, we chose either a small time T∗or small data in order to obtain �u�p X(T∗)≤r. Concretely, –for small time: Let ρ0and u0given. Taking r>2�u0�p W1,p(Ω), we chose T∗small enough in order to obtain the following inequality: CT ∗�r2+�ρ0�2p W2,p(Ω)exp �2CT p−1 p ∗r��≤r/2. –for small data: Now, the time Tis fixed. Then, we choose rsmall enough such that CTr 2≤r/2, and �u0�W1,p(Ω),�ρ0�W2,p(Ω)small enough verifying: �u0�p W1,p(Ω)+CT�ρ0�2p W2,p(Ω)exp �2CTp−1 pr�≤r/2. 3) Ris continuous from E into E: It suffices to prove sequential continuity, i. e. , if ¯ un→¯ uin E, then R(¯ un)→R(¯ u)inE. Let (¯ un)⊆Arsuch that ¯ un→¯ uin E.SinceAris a closed bounded set of X(T) (and X(T) is a reflexive Banach space), ¯ u∈Arand ¯ un�¯ uin X(T). We consider un=R(¯ un) and u=R(¯ u). In particular, (un)⊆Ar,u∈Arand there exists a subsequence (unk)⊂(un) and a limit function Υ∈Arsuch that: unk→Υstrongly in Eand weakly in X. (9) It suffices to prove that Υ=u,beinguthe unique solution of system (7), because in this case the whole sequence (un) converges to u. To this aim, we need to consider ρnthe solution of (8) associated to the velocity ¯ un. Then, using (3), one has that ρnis bounded in L∞(0,T;W2,p(Ω)) (and ∂tρnis bounded 5 in Lp(0,T;W1,p(Ω))), hence (ρn) is relatively compact in C([0,T]; W1,p(Ω)). Therefore, there exists a subsequence (ρnk)⊂(ρn) such that: ρnk→χstrongly in C([0,T]; W1,p(Ω)) and weakly-�in L∞(0,T;W2,p(Ω)). (10) Then, taking the limit in the equation ∂tρnk+( ¯ unk·∇)ρnk=0in(0,T)×Ω, we obtain that χ=ρ, where ρis the solution of (8) for the velocity ¯ u. Since this solution is unique, the previous convergence holds for the whole sequence (ρn). The next step is to take the limit in the system: ∂tun−ν∆un+∇π=−(¯ un·∇)¯ un−k∇ρn∆ρn,∇·un=0in (0,T)×Ω.(11) Then, thanks to the convergences for ¯ un, (10) for the whole sequence (ρn) and χ=ρand (11), one has that Υis a solution of (7). Since the solution of system (7) is unique, we conclude that Υ=uand the convergence (9) holds for the whole sequence (un). Appendix: Proof of inequality (5) Observe that �∇u�Lp(Ω)is a norm equivalent to �u�W1,p(Ω)in W1,p(Ω)∩V, and that: 1 p d dt �∇u�p Lp(Ω)=�Ω∇(∂tu)·�|∇u|p−2∇u�dΩ.Thus, integrating by parts on the right hand side: d dt�∇u�p Lp(Ω)≤C� Ω |∇u|p−2|D2u||∂tu|dΩ≤C�∇u�p−2 Lp(Ω)�D2u�Lp(Ω)�∂tu�Lp(Ω) ≤C�u�p−1 W2,p(Ω)�∂tu�Lp(Ω)≤C��u�p W2,p(Ω)+�∂tu�p Lp(Ω)� Therefore, for each t>0, �u�p L∞ t(W1,p(Ω)≤�u0�p W1,p(Ω)+C��∂tu�p Lp t(Lp(Ω)) +�u�p Lp t(W2,p(Ω))�. Remark 1 It is possible to make similar estimates in a different space of strong solutions (see [1] for the case of an Oldroyd model), obtaining the following regularity for s∈(1,+∞)and r∈(3,+∞): u∈Ls(0,T;W2,r(Ω)),∂tu∈Ls(0,T;Lr(Ω)),ρ∈L∞(0,T;W2,r(Ω)),∂tρ∈Ls(0,T;W1,r(Ω)). In this case, we use Giga-Sohr’s results [2] are used instead of those of Ladyzhenskaya-Solonnikov [5]. References [1] E. Fern´andez-Cara, F. Guill´en-Gonz´alez & R. R. Ortega, Some theoretical resuls concerning non newtonian fluids of the Oldroyd kind. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) Vol. XXVI (1998), 1-29. [2] Y. Giga & H. Sohr, Abstract Lpestimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102 (1991), 72-94. [3] D. J. Korteweg Sur la forme que prennent les ´equations du mouvement des fluids si l’on tient compte des forces capillaires caus´es par les variations de densit´e [on the form the equations of motions of fluids assume if account is taken of the capillary forces caused by density variations]. Archives N´eerlandaises des Sciences Exactes et naturelles, Series II, volume 6 (1901), 1-24. [4] I. Kostin, M. Marion, R. Texier-Picard & V. A. Volpert, Modelling of miscible liquids with the Korteweg stress, M2AN, Vol. 37, no5 (2003), 741-753. [5] O. A. Ladyzhenskaya & V. A. Solonnikov, Unique solvability of an initial and boundary problem for viscous incompressible nonhomogeneous fluids. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 52, pp. 52-109, 1975. [6] J. Simon, Compact sets in the space Lp(0,T;B). Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. [7] V. A. Solonnikov, Estimates of the solutions of the nonstationary Navier-Stokes systems, Tr. Mat. Inst. Akad. Nauk SSSR, 70, 213-317 (1964). Translated in Am. Math. Soc. Transl. Series 2, Volume 75 (1968). 6