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Short courses (invited speakers):

• Raphaël Danchin (LAMA, Paris-Est): Fourier analysis methods for compressible flows.
  
  Abstract

  Since the 80's, Fourier analysis methods have known a growing interest in the study of linear and nonlinear PDE's. In particular, techniques based on Littlewood-Paley decomposition and paradifferential calculus have proved to be very efficient for solving such equations in the whole space or the torus.

  In this course, we aim at giving a survey of how those techniques may be used for solving systems of equations arising in fluid mechanics. We shall in particular focus on the study of the Cauchy problem and long-time asymptotics for viscous or capillary compressible flows.

  
  
  
  
• Benoît Desjardins (ENS Paris): Global weak solutions for degenerate compressible Navier-Stokes systems.
  Abstract

  The objective of this course is to build global weak solutions for Navier-Stokes governing compressible viscous fluids with density dependent viscosities. These solutions have the property to satisfy energy inequalities corresponding to the physical energy of the system. The method introduced by P.--L. Lions (1993) and E. Feireisl (2000) to prove existence of global weak solutions with constant viscosities does not work with density dependent viscosities.

  In this case, the introduction of a mathematical entropy which is linked to a velocity associated to the gradient of the density allows to prove the existence of global weak solutions (D. Bresch, B.D., 2007). This method applies also to Saint-Venant for shallow water flows (degenerate viscosities depending on the height of the fluids) and to compressible diphasic models (models of mixture with two velocities with pressure equilibrium).

  We discuss recent results and difficulties associated to such models. We will also focus on compressible Navier-Stokes equations with heat conductivity describing recent results and open problems.

  
  
  
  
• Antonin Novotny (ANAM, Toulon): Compressible Navier-Stokes equations: existence of weak solutions, relative entropies and weak-strong uniqueness.
  Abstract

  In the first part of the minicourse we shall explain the main ideas and techniques behind the proof of existence of weak solutions to the system of partial differential equations describing the compressible fluids in barotropic regime, including effective viscous fluxes, oscillation defect measures and renormalized solutions to the continuity equation.

  In the second part we shall introduce the notion of relative entropies for this sytem and investigate the weak strong uniqueness principle as well as some of its consequences. Finally, the third part will be devoted to the generalization of some of the above concepts to the complete Navier-Stokes-Fourier system.

  
  
  
  
• Mikhail Perepelitsa (University of Houston, Houston): Solutions of the Navier-Stokes equations for compressible flow with discontinuous initial data.
  Abstract

  In this series of talks we will develop the existence theory for weak solutions of the Navier-Stokes equations for compressible flow in the intermediate regularity class. The solutions from this class enjoy a higher regularity compared to weak, energy solutions of the P.L.Lions-Feireisl theory, but at the same time they allow for discontinuities in the density and the gradient of the velocity. A typical problem modeled by such solutions is a motion of a blob (or a patch, for problems in $\mathbb{R}2$) of a higher density gas in a lower density background.

  We will start by considering the Cauchy Problem in $\mathbb{R}^n,$ $n{}={}2,3,$ and then discuss the specifics of the boundary value problems in a half-space $\mathbb{R}^n_+,$ with the Dirichlet and Navier boundary conditions for velocity.

  Following that we will focus on the dynamics of the interface of a density discontinuity in compressible viscous flows. We will establish that the smooth interface of a density discontinuity retains its structure during the motion, while the interface with an isolated corner singularity, generically, evolves into a cusp at the point of the singularity. We will point out certain analogies between this problem and the problem of the dynamics of vortex patches in 2-dimensional incompressible flows described by the Euler equations and, in particular, we will show the occurrence of the same phenomenon of the cusp formation from a corner singularity.

  
  
  
  

Forty minutes lectures:

• Sylvie Benzoni (ICJ, Lyon): Navier-Stokes in the Earth mantle.
  Abstract

  Our planet is known to have a liquid core surrounded by a mantle that behaves as an elastic material over human time scales, and is being convected as a highly viscous fluid over geological time scales. The talk will be concerned with the latter behaviour. The idea is to describe roughly what is known by geophysicists, and to raise a number of questions that can be addressed by mathematicians, in particular those having to do with small parameters --Reynolds number is believed to be of the order of 10^(-16)-- and phase transitions occurring far beyond depths that can be reached by drilling.

• Pierre Degond (IMT-CNRS, Toulouse): Constrained Navier-Stokes-like models in swarm dynamics.
  Abstract

  Swarm dynamics becomes increasingly important in biology and social sciences. Swarms consist of a large number of self-propelled interacting particles (such as e.g. bird flocks, fish schools, swimming bacteria or crowds). Swarm dynamics also provides a model for certain social phenomena such as consensus formation, mimetic behavior, bubble formation so on. In this talk, we focus on macroscopic models that can be derived from self-propelled particles interacting through alignment. These models are of the form of non-standard compressible Navier-Stokes like models which involve a geometrical constraint on the velocity. We will review how they can be derived and what we know (and do not know) about these models.

• Eduard Feireisl (Institute of Mathematics, Praha, Czech Republic): Inviscid incompressible limits of the full Navier-Stokes-Fourier system.
  Abstract

  We consider the full Navier-Stokes-Fourier system in the singular limit for the small Mach and large Reynolds and Peclet numbers, supplemented with ill prepared initial data on in the physical space $R^3$. The Euler-Boussinesq approximation is identified as the limit system. Our approach is based on the concept of relative entropy and the weak formulation of the Navier-Stokes-Fourier system.

• Emmanuel Grenier (UMPA, Lyon): Formal analysis of incompressible limit for inhomogeneous fluids.
  Abstract

  We will first discuss the incompressible limit of homonogeneous fluids and describe how to deal with the limit incompressible part and the highly oscillatory part composed of sound waves. We will then turn to the more difficult case of inhomogeneous fluids and detail its formal analysis. This is a joint work with D. Bresch and B. Desjardins.

• Boris Haspot (CEREMADE, Paris Dauphine): Convergence of capillary fluid models: from the non-local to the local Korteweg model.
  Abstract

  In a first part, we shall give new results of global strong solution for the Korteweg system. In a second part we would like to explain some links between the non-local and the local Korteweg model, and between the local Korteweg system and the compressible Euler equation.

• Aurélien Klak (LATP, Marseille): On the production of dissipation by interactions of oscillating waves in fluid mechanics.
  Abstract

  We introduce a two dimensional Navier-Stokes type model, forced by some polarized oscillating forces. We exhibit a family of solutions which time of existence is independent from the various scales involved in the equations. The family of solutions can asymptotically be described thanks to a boundary layer at time t=0. Furthermore, the interactions of oscillating waves macroscopically produce some viscosity. Finally, we perform some L²-type estimates to justify the expansion. The equilibrium between the hyperbolic singular part of the equations and the parabolic part has to be studied with care.

• Philippe LeFloch (LJLL-CNRS, Paris 6): Compressible fluid flows with finite energy
  Abstract

  I will talk about the Euler equations for compressible fluid flows in one space dimension, when the (real) fluid is governed by a general equation of state, and will discuss new existence and compactness results for weak solutions satisfying entropy inequalities. A general formula for the mathematical entropy pairs associated with the Euler equations is established, by introducing a fundamental kernel which is only H\"older continuous on the boundary of its support.

• Emmanuel Maitre or Stéphane Labbé (LJK, Grenoble): About a free boundary problem for Korteweg fluids.
  Abstract

  Lions and Masmoudi studied a decade ago the limit of compressible isentropic Navier-Stokes equations when the pressure law power goes to infinity. They obtained as limit a free boundary model between an incompressible fluid and a compressible one. Around the same years, Bresch, Desjardins and Lin showed existence of solution to the compressible Navier-Stokes equations with Korteweg term. The aim of the present work is to consider the limit toward the same kind of free boundary problem but with this Korteweg term.

• Alexander Mamontov(LIH, Novosibirsk, Russia): Existence theorems for the equations of multidimensional steady motions of two-component mixtures of viscous compressible fluids.
  Abstract

  We consider several versions of the model mentioned in the title: heat-conductive or not, with different additional assumptions. Flows in a bounded domain are investigated. As a result, we obtain global existence theorems for all models under consideration in the class of weak solutions.

• Pascal Noble (ICJ, Lyon): Falling films down an incline plane: modeling and long-time behavior.
  Abstract

  We consider an incompressible viscous fluid flowing down an incline plane and modeled by free surface Navier-Stokes equations. It is well known since Yih that for a given inclination, the steady states are unstable above a critical discharge Reynolds number (or equivalently above a critical discharge rate) and zero wavenumber perturbations are the most unstable. In this situation, hydrodynamic instabilities with a small wavenumber, so called roll-waves, arise. In this talk, I will show how to model such a situation. Indeed, this type of flow corresponds to a shallow water regime. We will present a shallow water model that is "consistent" with the Navier-Stokes equations and show a "convergence" result when steady states are stable. The situation is trickier in the unstable case: here we consider directly the shallow water equations and study the stability of roll-waves that are periodic travelling waves solutions of the shallow water equations. We provide here the first spectral/nonlinear stability result for this kind of waves in the context of conservation laws.

• Denis Serre (UMPA, Lyon): Multi-dimensional rarefaction waves.
  Abstract

  We consider self-similar solutions of systems of conservation laws (Riemann problem). In several space dimensions, the Lipschitz solutions are more complicated than the rarefaction waves of the one-D case. We show that the jump of the gradient can be calculate explicitly; a strange phenomenon appears in three space dimensions. We also show that a singular limit occurs in the so-called Regular Reflection of a gas against a wall, as the reflection tends to be normal.

• Raafat Talhouk (Faculte des Sciences, Liban): Slightly compressible viscoelastic fluid flows and the incompressible limit..
  Abstract

  In this lecture we introduce the Oldroyd model for slightly compressible viscoelastic fluids. We show local existence (and uniqueness) of regular flows in a bounded domain by a classical fixed point argument. Using some a priori estimates which are uniform in the Mach number, we prove that the solutions are global in time. We also show that slightly compressible flows with well-prepared initial data converge to incompressible ones when the Mach number converges to zero.