Analysis of free boundary problems and compressible fluids equations

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There are two facets to my research programme; Both deal with the mathematical analysis of nonlinear partial differential equations arising from the modeling of natural phenomena.The first one is motivated by material sciences and is concerned with free boundary problems, which typically describe the evolutions of interfaces between different materials or different phases of a same material (such as the melting of a block of ice in a glass of water). The mathematical analysis of such models is very delicate and there are many open questions. The existence, regularity and asymptotic behaviour of the solutions are still considered to be subtle issues in many cases. One of the goal of this proposal is to study the effects of small perturbations in the properties of one of the materials (due, for example, to the chemical contamination  of a surface) on the global behaviour of the solutions. I use some classical techniques of the theory of homogenization, but the main tools are specific to this framework. Note that despite considerable developments in the theory of homogenization over the last 25 years, the homogenization of free boundary problems is still poorly understood.The second part of my research programme deals with the notoriously rich and challenging problems of fluid dynamics. I am interested in existence and regularity issues for compressible fluids equations and, in particular, the compressible Navier-Stokes equations. The main goal of my proposal is to study how the behavior of the viscosity coefficients (which are often assumed to be constant, but depend on the temperature of the fluid in many physical regimes) might affect the properties of the system of equations. Here, I use many classical tools (such as entropy inequalities) but also some very old techniques from the Calculus of Variations whose applications in fluid dynamics have so far been overlooked.