接触Hamilton系统与一类偏微分方程粘性解的奇性传播

The notion of viscosity solutions was introduced by Crandall and Lions in 1983, which is a kind of weak solutions of partial differential equations. The points at which the viscosity solution fails to be differentiable are called singular points. The problem of propagation of singularities is always highly focused on. The project focuses on the problem of propagation of singularities of viscosity solutions of a class of first-order partial differential equations (GHJ). The characteristic equations for this class of partial differential equations are contact Hamiltonian systems. Our method originates from dynamical systems. More precisely, we will discuss the following three problems: 1. the necessary and sufficient conditions for (global) propagation of singularities of viscosity solutions of equation (GHJ); 2. the topology of the set of singularities of viscosity solutions of equation (GHJ); 3. the relationship between propagations of singularities for different viscosity solutions of equation (GHJ) and the problem of propagation of singularities along surfaces.

粘性解是由Crandall与Lions于1983年引入的一种偏微分方程弱解的概念。粘性解的不可微点称为奇点，奇点的传播问题一直备受关注。本项目旨在研究一类一阶偏微分方程（GHJ）粘性解的奇性传播问题。这类偏微分方程的特征线方程是接触Hamilton系统。我们的方法来源于动力系统。具体地，本项目将围绕以下三个问题进行研究：1.方程（GHJ）粘性解奇性（全局）传播的充分必要条件；2.方程（GHJ）粘性解奇点集的拓扑结构；3.方程（GHJ）的不同粘性解的奇性传播之间的关系以及奇性传播的曲面问题。

项目组成员与合作者得到了切触哈密顿系统Mather理论和弱KAM理论的部分结果，并应用这些动力学结果在适当条件下，解决了切触型哈密顿-雅可比方程粘性解的适定性、长期行为等问题；对哈密顿-雅可比方程粘性解的Dirichlet问题进行了研究，用动力学方法刻画了奇性传播规律；将动力学方法带入了平均场博弈系统（哈密顿-雅可比方程与连续性方程构成的正倒向方程组）数学模型的理论分析。上述成果发表在CMP, JMPA, CPDE, JDE, JDDE, PJM, DCDS, DGA, JDDE。