Analysis of free boundary problems and compressible fluids equations

我的研究计划有两个方面；两者都处理由自然现象建模引起的非线性偏微分方程的数学分析。第一个是由材料科学推动的，与自由边界问题有关，它通常描述不同材料之间或同一材料的不同阶段之间界面的演变（例如一块冰在一杯水中的融化）。这类模型的数学分析非常精细，有许多悬而未决的问题。在许多情况下，解的存在性、正则性和渐近性仍然被认为是一个微妙的问题。本提案的目标之一是研究其中一种材料性质的微小扰动（例如，由于表面的化学污染）对溶液整体行为的影响。我使用了一些同质化理论的经典技术，但主要的工具是针对这个框架的。请注意，尽管均匀化理论在过去25年中取得了相当大的发展，但自由边界问题的均匀化仍然知之甚少。我的研究计划的第二部分涉及流体动力学中众所周知的丰富和具有挑战性的问题。我对可压缩流体方程的存在性和规律性问题很感兴趣，特别是可压缩的Navier-Stokes方程。我建议的主要目标是研究粘度系数（通常被认为是恒定的，但在许多物理制度中取决于流体的温度）的行为如何影响方程组的性质。在这里，我使用了许多经典工具（如熵不等式），但也使用了一些非常古老的技术，这些技术来自于变分学，它们在流体动力学中的应用迄今为止一直被忽视。

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.