Analysis of free boundary problems and compressible fluids equations

自由边界问题和可压缩流体方程分析

• 批准号: 341253-2007
• 负责人: Mellet, Antoine
• 金额: $ 1.6万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=365839
• 关键词: Analysis; free; boundary; problems; compressible

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.

我的研究计划有两个方面：这两门学科都是对自然现象建模过程中产生的非线性偏微分方程进行数学分析;第一门学科受材料科学的启发，关注自由边界问题，它通常描述不同材料或同一材料的不同相之间的界面演化（如一杯水中冰块的融化）。这种模型的数学分析是非常微妙的，有许多悬而未决的问题。解的存在性、正则性和渐近性态在许多情况下仍然被认为是微妙的问题。该提案的目标之一是研究其中一种材料的性质的小扰动（例如，由于表面的化学污染）对解决方案的全局行为的影响。我使用了同质化理论的一些经典技术，但主要工具是特定于这个框架的。请注意，尽管在过去的25年中，均匀化理论有了相当大的发展，但对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众所周知的流体动力学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正则性问题感兴趣，特别是可压缩Navier-Stokes方程。我的建议的主要目标是研究粘性系数（通常假设为常数，但在许多物理状态下取决于流体的温度）的行为如何影响方程组的性质。在这里，我使用了许多经典的工具（如熵不等式），但也使用了一些来自变分法的非常古老的技术，这些技术在流体动力学中的应用迄今为止一直被忽视。