Free Boundary Problems and Mass Transfer

摘要--------------------------------------------------本课题包括两个主要课题：(1)非线性椭圆方程的自由边界问题。目的之一是发展研究可压缩流体动力学模型中自由边界问题的技术，特别是研究稳定和自相似势流的欧拉方程的多维跨音速激波的存在性、唯一性和稳定性。如果流动是稳定的或自相似的，欧拉方程可以写成速度势的单二阶非线性混合椭圆双曲方程。跨音速激波在解的梯度上是不连续的，使得方程的类型在激波表面上从双曲型变为椭圆型。跨音速激波出现在许多重要的物理情况下（围绕障碍物的稳定超音速流动，自相似流动的激波反射）。(2) Monge-Kantorovich传质问题。要研究的问题包括解的几何和测量理论性质，以及在偏微分方程中的应用。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。然而，对于非线性偏微分方程，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，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。本论文拟开展最优运输问题的理论与应用研究。

PI: Mikhail Feldman, University of WisconsinDMS-0200644ABSTRACT--------------------------------------------------The project consists of two main topics: (1) Free boundaryproblems for nonlinear elliptic equations. One of objectivesis to develop techniques for studying free boundary problemsarising in the models of compressible fluid dynamics, inparticular to study existence, uniqueness and stability ofmultidimensional transonic shocks for Euler equations forsteady and self-similar potential flows. Euler equations canbe written as a single second order, nonlinear elliptic-hyperbolicequation of mixed type for the velocity potential, if the flowis steady or self-similar. Transonic shocks are discontinuitiesin the gradient of the solution such that the type of equationchanges from hyperbolic to elliptic across the shock surface.Transonic shocks arise in many situations of physical importance(steady supersonic flows around an obstacle, shock reflection forself-similar flows). (2) Monge-Kantorovich mass transfer problem.The questions to study include geometric and measure-theoreticproperties of solutions, and applications to partial differentialequations.Free boundary problems arise naturally in many models in physics,fluid dynamics, economics. Free boundaries correspond to sharpchanges in the variables describing the problem. Significantprogress has been made during last several decades in the studyof free boundary problems. However in the case of nonlinear partialdifferential equations many important questions are yet to be studied.This is the first theme of the project. Better understanding ofproperties of free boundaries, such as stability, makes possible tounderstand complex phenomena in models and applications. We plan tostudy 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 ofthis problem within and beyond mathematics were discovered, inparticular its connections to nonlinear partial differentialequations, and applications in models for front formation in theatmosphere, kinetic theory, fluid flow, elastic crystals, granularmaterials, and microeconomic decision problems. We plan to workon theory and applications of optimal transportation problem.