Nonlinear Partial Differential Equations

非线性偏微分方程项目摘要lawrence Craig evans这个项目是分析一类非线性偏微分方程（PDE）解的性质。一类是“弱KAM”理论，它研究非可积哈密顿系统，使用变分和PDE方法来识别与动作最小化措施相关的某些类别的良好行为轨迹。新的PDE方法将试图发现，例如，非共振条件的全部含义。另一类问题是无穷拉普拉斯方程和Aronsson方程解的正则性问题。这些高度退化的、高度非线性的方程对超范数中的变分问题以及某些随机回合随机对策进行了建模。这里的主要目标是证明弱解是连续可微的。该项目还将研究“湖泊和河流”PDE模型。这是一个非线性抛物扩散方程的奇异极限，它有一个简单的解释：极限解要么拥抱一个给定的景观，要么形成由“河流”连接的平坦“湖泊”。技术上的挑战将是开发PDE和测量理论工具来理解和预测“湖泊和河流”的位置。本研究计划也将探讨使用某些熵型不等式来研究正则化非线性扩散方程的渐近极限。我们期望对于非单调线性，这个渐近极限会产生滞后效应，称为动态相变。一个不同的和相当投机的工作是为完全非线性偏微分方程发展某种类似于守恒定律的重要动力学公式。这些不同的PDE出现在各种环境中，包括物理，博弈论和图像处理，因此结果在许多应用领域都很重要。

NONLINEAR PARTIAL DIFFERENTIAL EQUATIONSProject AbstractLawrence Craig EvansThis project is to analyze the properties of solutions of certain classes of nonlinear partial differential equations (PDE). One class is "Weak KAM" theory which studies far-from-integrable Hamiltonian systems, using variational and PDE methods to identify certain classes of well-behaved trajectories, associated with action minimizing measures. New PDE approaches will try to discover, for instance, the full implications of nonresonance conditions.Another class of problems is the regularity of solutions of the infinity Laplacian equation and Aronsson's equation. These highly degenerate, highly nonlinear equations model variational problems in the sup-norm as well as certain random-turn stochastic games. A major goal here will be to prove that weak solutions are continuously differentiable.The project will also investigate a "lakes and rivers" PDE model. This is a singular limit of a certain nonlinear parabolic diffusion equation, which has a simple interpretation: the limiting solution either hugs a given landscape or else forms flat ``lakes'' connected by ``rivers''. The technical challenge will be to develop PDE and measure theory tools to understand and predict the location of the "lakes and rivers".This research project will also explore the use of certain entropy type inequalities to study the asymptotic limit of a regularized nonlinear diffusion equation. We expect that for a nonmonotone linearity this asymptotic limit should develop hysteresis effects, called dynamic phase transitions. A different and quite speculative undertaking is to develop for fully nonlinear PDE some kind of analog of the important kinetic formulation for conservation laws.These various PDE arise in a variety of settings including physics, game theory and image processing, so the results should be of importance in many application areas.