This project is to continue working on some asymptotic problems in Fluid Mechanics, Gas Dynamics and Plasma physics, as well as to study some well-posedness questions arising in free boundary problems and non-Newtonian fluids. These asymptotic problems arise when a dimensionless parameter (such as the Reynolds number or the ratio between the electron thermal speed and the speed of light) goes to zero or to infinity. In studying these problems, many mathematical difficulties arise. These difficulties are mainly due to the change of the type of the equations, the presence of many temporal and spatial scales, the presence of resonances, the presence of boundary layers, ... . Many tools have been developed to circumvent these difficulties such as the introduction of different types of measures to describe the defect of strong convergence, the use of compensated compactness type arguments, the use of averaging lemma, the use of energy methods and relative entropy methods, ... . New tools will be developed in this work. Hopefully, they can be applied to other areas of mathematics.
The study of these problems is very important to get a better understanding about the behavior of complicated systems in different limiting cases. This also allows an improved understanding of the real physical phenomena taking place. It also gives a better knowledge about the domain of validity of each simplifying model. This is very important for engineers and physicists who are looking for the simplest model that captures the phenomena to implement numerically or to apply in real life. One of these problems is the hydrodynamic limit of the Boltzmann equation when the Knudsen number goes to 0 especially in bounded domains. Formally, the limit equation (Euler system, Navier-Stokes, ...) depends on additional parameters such as the Mach number, the Reynolds number and the time scaling. In particular, one of the goals of the project is to understand fluid boundary conditions from kinetic ones. This is a step towards understanding the coupling at the boundary between fluid equations and kinetic equations. Of course this is a very important question for numerical simulations when we have to go from a domain where a kinetic model is being used to a domain where a fluid model is being used. Besides, another goal of this project is the study of some non-Newtonian fluids and especially polymeric liquids (egg white, blood or dough for example). These systems require a coupling between fluids and polymers and are of great interest in many branches of applied physics, chemistry and biology.
Many fundamental problems in Fluid Mechanics and Plasma Physics were solved. With Levermore, we gave a full justification of the derivation of fluid equations (such as the Navier-Stokes system) from kinetic equations (Boltzmann equation). This is a fundamental questions that was formulated by Hilbert in his sixth problem in 1900. A fundamental step was done by L. Saint-Raymond and F. Golse. With a former student Arsenio, we develop new regularity estimates for renormalized solutions of the Boltzmann equation with long-range interactions which are used by Arsenio in his thesis to prove the hydrodynamic limit for potential without cut-off. With Ambrose, we gave an existence result for 3D vortex sheets with surface tension. We use a special coordinate system, namely conformal coordinates to parametrize the surface. Our parametrization has the advantage of being suitable for numeric simulations since we reduce a 3D problem to a simple problem on a surface. With Nakanishi, we study the convergence in the energy space of solutions to the Klein-Gordon-Zakharov system towards a solution of coupled nonlinear Schrodinger equations when two dimensionless parameters $c$ and $alpha$ go to infinity. These two parameters are related to the plasma frequency and the plasma sound speed. With Wong, we construct solutions in Sobolev spaces to the hydrostatic Euler equations and to the Prandtly system under some monotonicity assumption. These two models are very important models to understand the inviscid limit (when the viscosity tends to zero). Also, with Rousset we prove the convergence of solutions to the Navier-Stokes system with Navier boundary condition towards a solution to the Euler system when the viscosity goes to zero. The novelty of this work is that we are able to prove uniform bounds in conormal Sobolev spaces without constructing a boundary layer. With Juhi Jang, we prove well-posedness for the free-boundary compressible Euler equations with physical vacuum in one dimension. We also extended this to the 3D case using Lagrangian coordinates and studying a degenerate acoustic system. Similar results where obtained by Coutand and Schkoller. Also, we gave a rigorous derivation of the anelastic approximation starting from the compressible Navier-Stokes system with gravitational potential when the Mach number and the Froude number go to zero with the same speed. All these results have important applications from physical point of view
