Free Boundary Problems, Mass Transfer and Nonlinear Dynamics

项目摘要mikhail feldman该项目主要包括两个主题：(1)可压缩流体动力学模型中出现的自由边界问题。目的是研究稳态和自相似势流欧拉方程多维跨音速激波的存在性、唯一性和稳定性。欧拉方程由质量守恒定律和速度伯努利定律组成，当流体稳定或自相似时，欧拉方程的速度势可以写成二阶非线性椭圆-双曲方程。跨音速激波是溶液梯度中的不连续点，使得方程的类型在激波表面上从双曲型变为椭圆型。跨音速冲击出现在许多重要的物理场合。跨音速激波解的边值问题可以表示为椭圆相的自由边值问题。该框架将应用于研究超声速流过锥形体和光滑凸障碍物时的自相似激波反射和稳定跨声速激波等几个重要情况。这些问题也将在完全可压缩欧拉系统的更一般的框架中考虑。(2)另一个研究领域是Monge-Kantorovich传质问题。要研究的问题包括最优映射的几何和测量理论性质，以及在物理模型中出现的偏微分方程的应用。这包括半匀转方程系统，一个大尺度大气/海洋流动的模型。另一个领域是研究海森堡群上的蒙格问题。自由边界问题在物理学、流体动力学、经济学的许多模型中自然出现。自由边界对应于描述问题的变量的急剧变化。近几十年来，自由边界问题的研究取得了重大进展。但是对于非线性偏微分方程，特别是混合型偏微分方程，还有许多重要的问题有待研究。这是该项目的第一个主题。更好地理解自由边界的性质，例如稳定性，使理解模型和应用中的复杂现象成为可能。我们计划研究可压缩流体或气体流动中的跨音速激波。项目的另一个领域是最优运输问题。最近，这个问题在数学内外的许多基本性质和重要应用被发现，特别是它与非线性偏微分方程的联系，以及在大气锋面形成模型、动力学理论、流体流动、弹性晶体、颗粒材料和微观经济决策问题中的应用。我们计划研究最优运输问题的理论和应用。该项目产生的更广泛的影响将是分析工程和气象学中的一些重要问题。

''Free boundary problems, mass transfer and nonlinear dynamics''Project AbstractMikhail FeldmanThe project consists of two main topics: (1) Free boundary problems arising in the models of compressible fluid dynamics. The objective is to study existence, uniqueness and stability of multidimensional transonic shocks for Euler equations for steady and self-similar potential flows. Euler equations, consisting of the conservation law of mass and Bernoulli law for the velocity, can be written as a second-order, nonlinear elliptic-hyperbolic equation for the velocity potential, if the flow is steady or self-similar. Transonic shocks are discontinuities in the gradient of a solution, such that the type of equation changes from hyperbolic to elliptic across the shock surface. Transonic shocks arise in many situations of physical importance. Boundary value problems for transonic shock solutions can be formulated as a free boundary problem for the elliptic phase. This framework will be applied to study several important situations, including self-similar shock reflection, and steady transonic shocks in supersonic flow past conical body and smooth convex obstacle. These problems will also be considered in the more general framework of full compressible Euler system.(2) Another area of research is Monge-Kantorovich mass transfer problem. The questions to study include geometric and measure-theoretic properties of optimal maps, and applications to partial differential equations arising in physical models. This includes the system of semigeostrophic equations, a model of large-scale atmosphere/ocean flows. Another area is to study Monge problem on Heisenberg group.Free boundary problems arise naturally in many models in physics, fluid dynamics, economics. Free boundaries correspond to sharp changes in the variables describing the problem. Significant progress has been made during last several decades in the study of free boundary problems. However in the case of nonlinear partial differential equations and especially equations of mixed type many important questions are yet to be studied. This is the first theme of the project. Better understanding of properties of free boundaries, such as stability, makes possible to understand complex phenomena in models and applications. We plan to study transonic shocks in a flow of compressible fluid or gas. Another area of the project is optimal transportation problem. Recently many fundamental properties and important applications of this problem within and beyond mathematics were discovered, in particular its connections to nonlinear partial differential equations, and applications in models for front formation in the atmosphere, kinetic theory, fluid flow, elastic crystals, granular materials, and microeconomic decision problems. We plan to work on theory and applications of optimal transportation problem. A broader impact resulting of the project will be the analysis of some important problems in engineering and meteorology.