Nonlinear Partial Differential Equations

项目摘要劳伦斯·克雷格·埃文斯这个项目是分析某些类型的非线性偏微分方程解的性质。其中一类是“弱KAM”理论，它研究远非可积的哈密顿系统，使用变分和偏微分方程方法来识别某些类行为良好的轨迹，并与行动最小化措施相关联。例如，新的偏微分方程法将试图发现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。这些高度退化的、高度非线性的方程在超范数下模拟变分问题，以及某些随机转向随机对策。这里的一个主要目标是证明弱解是连续可微的。该项目还将研究一个“湖泊和河流”的PDE模型。这是一类非线性抛物型扩散方程的奇异极限，它有一个简单的解释：极限解要么拥抱给定的景观，要么形成由河流连接的平坦的湖泊。技术挑战将是开发偏微分方程和测量理论工具来理解和预测“湖泊和河流”的位置。本研究项目还将探索使用某些熵型不等式来研究正则化的非线性扩散方程的渐近极限。我们预计，对于非单调线性，这个渐近极限应该会产生滞后效应，称为动态相变。另一项不同且颇具投机性的工作是为完全非线性的偏微分方程组建立某种类似的守恒定律的重要动力学公式。这些不同的偏微分方程组出现在包括物理、博弈论和图像处理在内的各种环境中，因此其结果在许多应用领域都应该是重要的。

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.