Mathematical Sciences: Nonlinear Partial Differential Equations and Applications

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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.