Benoît Desjardins

On some compressible fluid models: Korteweg, lubrication, and shallow water systems

Abstract In this article, we give some mathematical results for an isothermal model of capillary compressible fluids derived by Dunn and Serrin in [1]Dunn JE, Serrin J. On the thermodynamics of interstitial working. Arch Rational Mech Anal. 1985; 88(2):95–133), which can be used as a phase transition model. We consider a periodic domain Ω = T d (d = 2 ou 3) or a strip domain Ω = (0,1) × T d −1. We look at the dependence of the viscosity μ and the capillarity coefficient κwith respect to the density ρ. Depending on the cases we consider, different results are obtained. We prove for instance for a viscosity μ(ρ) = νρ and a surface tension the global existence of weak solutions of the Korteweg system without smallness assumption on the data. This model includes a shallow water model and a lubrication model. We discuss the validity of the result for the shallow water equations since the density is less regular than in the Korteweg case.

INCOMPRESSIBLE LIMIT FOR SOLUTIONS OF THE ISENTROPIC NAVIER-STOKES EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS

We study here the limit of global weak solutions of the compressible Navier-Stokes equations (in the isentropic regime) in a bounded domain, with Dirichlet boundary conditions on the velocity, as the Mach number goes to 0. We show that the velocity converges weakly in L 2 to a global weak solution of the incompressible Navier-Stokes equations. Moreover, the convergence in L 2 is strong under some geometrical assumption on.

Recent Mathematical Results and Open Problems about Shallow Water Equations

The purpose of this work is to present recent mathematical results about the shallow water model. We will also mention related open problems of high mathematical interest.

Singular ordinary differential equations homogeneous of degree 0 near a codimension 2 set

This work deals with an example of class of ordinary differential equations which are singular near a codimension 2 set, with an homogeneous singularity of degree 0. Under some structural assumptions, we prove that for almost all initial data there exists a unique global solution and study the evolution of the Lebesgue measure of a transported set of initial data. The analysis is motivated by the Low Mach number asymptotics of compressible fluid models in the case of non isentropic flows, which involves such dynamical systems.

Multi-fluid Models Including Compressible Fluids

In this chapter, we focus on multi-fluid models including compressible fluids in the isentropic or isothermal case. The main objective of these notes is to present at the level of beginners an introduction to mesoscopic multi-fluid models. The guideline is to show how we derive mathematically some multi-fluid systems starting from mono-fluid models and how we use the mathematical structures of multi-fluid PDEs to prove well-posedness (local strong and global weak solutions). We hope by this chapter to motivate young researchers to work on such a difficult topic important for applications in industry (turbulent mixing in D. Bresch is partially supported by the ANR13-BS01-0003-01 project DYFICOLTI. M. Hillairet is partially supported by the ANR13-BS01-0003-01 project DYFICOLTI. D. Bresch ( ) LAMA UMR 5127 CNRS Batiment le Chablais, Université de Savoie Mont-Blanc, Le Bourget du Lac, France e-mail: didier.bresch@univ-smb.fr; didier.bresch@univ-savoie.fr B. Desjardins Fondation Mathématique Jacques Hadamard, CMLA, ENS Cachan, CNRS and Modélisation Mesures et Applications S.A., Paris, France e-mail: Benoit.Desjardins@mines.org J.-M. Ghidaglia CMLA, ENS Cachan, CNRS, Université Paris-Saclay, Cachan, France e-mail: jmg@cmla.ens-cachan.fr E. Grenier Unité de Mathématiques Pures et Appliquées, ENS Lyon, Lyon Cedex 07, France e-mail: egrenier@umpa.ens-lyon.fr; emmanuel.grenier@ens-lyon.fr M. Hillairet Institut Montpelliérain Alexander Grothendiek, UMR5149 CNRS, Université de Montpellier, Montpellier, France e-mail: matthieu.hillairet@univ-montp2.fr

On the Regularity of Solutions of the Compressible Isentropic Navier-Stokes Equations

Small time regularity of solutions of the compressible isentropic Navier-Stokes equations is investigated in dimension N = 2 or 3 under periodic boundary conditions. The initial density is not required to have a positive lower bound. We prove that weak solutions in T 2 remain smooth as long as the density is bounded in L ∞(T 2).

Oscillatory Limits with Changing Eigenvalues: A Formal Study

This paper deals with oscillatory limits with changing eigenvalues, more precisely with possibly crossing eigenvalues in space dimension greater than 1. The goal being to underline the various difficulties, to analyze them formally and present some related mathematical results obtained recently by the authors.

Existence of Compactly Supported Solutions for a Degenerate Nonlinear Parabolic Equation with NonLipschitz Source Term

In this paper the authors study the existence of nonnegative compactly supported solutions of a nonlinear degenerate parabolic equation with a non-Lipschitz source term in one space dimension. More precisely, they investigate the properties of nonnegative solutions of the problem {∂tu−ν∂2x(up)=uα+Ψ(u),u(0,x)=u0, x∈R, t>0, u=u(t,x),(1) where Ψ∈C∞ is an increasing function, satisfying Ψ(0)=0 (unforced case), Ψ(x)≤Cx for some C>0 and every x≥0. Equation (1) mimics the properties of the classical k-e system-model in the context of turbulent mixing flows with respect to nonlinearities and support properties of solutions. The authors highlight that the originality of the method resides in the fact that they deal with a non-Lipschitz source term, and in the comparison of not only the speed but also the acceleration of the boundary of the compact support. Here p>1 so that the diffusion is strongly degenerate and α>0 with 2−p≤α<1, so the nonlinearity is not Lipschitz. Equation (1) is a crude simplification of a k-e system of the form ∂tk+∂x(ku)+e=∂x(Cμk2e∂xk)+Pk2e, ∂te+∂x(eu)+C1e2k=∂x(Cμσek2e∂xe)+C2Pk, where k and e denote the specific turbulent kinetic energy and its dissipation rate, u is the velocity field, and P is proportional to turbulence production terms. The present work is focused on the lack of Lipschitz continuity of the source term and the degeneracy of the diffusion. Actually, the study is restricted to the scalar case, as in the k-l turbulence model, where l=k3/2/e is constant. Nonlinear parabolic equations similar to (1) appear also in various applications, most frequently to describe phenomena of thermal propagation in an absorptive medium. Note that such kinds of equations have been studied for slow or fast diffusion and strong or small absorption; see [R. Ferreira and J. L. Vazquez, Nonlinear Anal. 43 (2001), no. 8, Ser. A: Theory Methods, 943-985; MR1812069 (2002f:35141); R. Ferreira, V. A. Galaktionov and J. L. Vazquez, Nonlinear Anal. 50 (2002), no. 4, Ser. A: Theory Methods, 495-507; MR1923525 (2003h:35109); V. A. Galaktionov, S. I. Shmarev and J. L. Vazquez, Arch. Ration. Mech. Anal. 149 (1999), no. 3, 183-212; MR1726675 (2001k:35167)] and the references therein. Finally, the authors prove the following result: Theorem. Assume that u0 is a smooth nonnegative function with support of the form [α,β] and that the derivatives of up0 at α, β do not vanish. Then there exists a smooth, nonnegative, compactly supported solution u to (1) defined for all positive times.