Mathematical Sciences: Nonlinear Partial Differential Equations and Applications

这个项目将研究奇异摄动问题，完全非线性的一阶和二阶偏微分方程组和控制理论，力学(多孔介质方程)和非线性波。将考虑一大类与模拟物理、化学和生物现象的反应扩散方程或系统有关的奇异摄动问题，在这些问题中，锋面会自然发展很长一段时间。从PI引入的一种新方法的观点出发，研究一类与气体动力学和辐射传输方程有关的奇异摄动问题。内容包括完全非线性偏微分方程组及其与控制理论和微分对策的关系，与多孔介质方程理论有关的问题，流体中波的稳定性，以及非线性薛定谔方程的奇异性。这些研究在火焰传播、相变和种群进化等主题中得到了应用。

This project will study singular perturbation problems, fully nonlinear first-and-second-order pde's and control theory, mechanics (porous-medium equation), and nonlinear waves. A large class of singular perturbation problems related to reaction-diffusion equations or systems which model physical, chemical and biological phenomena, in which fronts develop naturally for large times will be considered. A class of singular perturbations related to gas kinetics and radiative transfer equations will be studied from the point of view of a new methodology introduced by the PI. Fully nonlinear pde's and their relations to control theory and differential games, questions related to the theory of porous-medium equations, the stability of waves in fluids, and the singularities of the nonlinear Schrodinger equation will be encompassed. These studies find applications in topics such as flame propagation, phase transitions, and evolution of populations.