Didier Bresch

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.

On the existence of solutions for non-stationary third-grade fluids

Abstract This paper is devoted to non-stationary third-grade fluids. We present some existence and uniqueness results in Lr where r>3 for open sets only of class C 2 . The proof is based on a fixed point method.

A vertical diffusion model for lakes

The motion of a fluid in a lake with small depth compared to width is investigated. We prove that when the depth goes to 0, the solution of the stationary Navier--Stokes equations with adherence at the bottom and traction by wind at the surface, once conveniently normalized, goes to a three-dimensional limit which is the solution of an incompressible model with vertical diffusion. The limit velocity is given in terms of the vertical coordinate and of the limit pressure. This pressure, which depends only on the horizontal coordinates, is driven by a two-dimensional equation on the surface degenerating on the shore, which is solved in a weighted space. Thus, a three-dimensional approximation is obtained by a simple two-dimensional computation.

A corrector for the sverdrup solution for a domain with islands

In this paper we look at the influence of the Coriolis force on the quasi-geostrophic equations on a domain with islands. We prove that asymptotically we obtain the solution of the Sverdrup equation with homogeneous Dirichlet conditions on the inward boundary plus a corrector function which takes into account the presence of the islands. This work is motivated by the fact that in oceanography most of the surfaces are not simply connected. This is the case for example for the North Pacific with the Japanese islands. At our knowledge, in all the previous mathematical works, just simply connected domains have been considered. Finally we will give some simple numerical simulations related to the Stommel model to see the importance of the corrector.

On the Effect of Friction on Wind Driven Shallow Lakes

Abstract. We investigate the steady motion of a liquid in a lake, modeled as a thin domain. We assume the motion is governed by Navier—Stokes equations, while a Robin-type traction condition, and a friction condition is prescribed at the surface and at the bottom, respectively. We also take into account Coriolis forces. We derive an asymptotic model as the aspect ratio

Multi-fluid Models Including Compressible Fluids

\delta

Newtonian Limit for Weakly Viscoelastic Fluid Flows

= depth/width of the domain goes to 0. When the Reynolds number is not too large, this is mathematically justified and the three-dimensional limit velocity is given in terms of wind, bathymetry, depth and of a two-dimensional potential. Numerical simulation is carried out and the influence of traction condition reading is experienced.

On wind driven geophysical flows without bottom friction

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

Existence of Global Weak Solutions for a 2D Viscous Shallow Water Equations and Convergence to the Quasi-Geostrophic Model

This article addresses the low Weissenberg asymptotic analysis (Newtonian limit) of some macroscopic models of viscoelastic fluid flows in the framework of global weak solutions. We investigate the convergence of the corotational Johnson--Segalman, the FENE-P (closure proposed by Peterlin of the Finitely Extensible Nonlinear Elastic model), and the Giesekus and Phan-Thien and Tannes models. Relying on a priori bounds coming from energy or free energy estimates, we first study the weak convergence toward the Navier--Stokes system. We then turn to the main focus of our paper, i.e., the strong convergence. The novelty of our work is to address these issues by relative entropy estimates, which require the introduction of some corrector terms. We also take into account the presence of defect measures in the initial data, uniform with respect to the Weissenberg number, and prove that they do not perturb the Newtonian limit of the corotational system.

On the uniqueness of weak solutions of the two-dimensional primitive equations

SuntoIn questo lavoro studiamo il moto di un fluido in un dominio sottile soggetto all’azione del vento in superficie ed a condizioni di slipping sul fondo. Viene derivato, dalle equazioni di Navier-Stokes in presenza della forza di Coriolis, un modello asintotico nel limite in cui il rapporto tra la profondità e la larghezza del dominio tende a zero, per numeri di Reynolds non troppo grandi.AbstractThe motion of a fluid in a thin domain subject to wind traction at the surface and to slipping at the bottom is investigated. An asymptotic model is derived from Navier-Stokes equations with Coriolis force as the aspect ratio δ=depth/width of the domain go to 0, for not too large Reynolds number.