Mathematical Sciences: Nonlinear Partial Differential Equations with Applications to Phase Transitions, Front Propagation and Mechanics

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Souganidis 9403412 Work supported by this grant covers a broad range of problems arising in the areas of hyperbolic and parabolic nonlinear partial differential equations. Three primary themes will be emphasized. First, work will be done on the kinetic theory of conservation laws. The goal is to obtain new regularity and compactness results for systems of conservationlaws, using a kinetic type formulation/approximation for these systems. Second, studies will be made on first- and second-order fully nonlinear degenerate elliptic and parabolic equations. The emphasis is on the analysis of discontinuous control problems, on proving convergence, on obtaining error estimates for numerical schemes and finally, on studying several issues regarding the regularity and uniqueness of weak solutions. Lastly, work will be done on generalized front propagation and phase transitions. Much of the focus will be on studying the regularity of moving fronts past their singularities as well as the theory of phase transitions. The latter includes singular limits of reaction-diffusion equations and systems, macroscopic (hydrodynamic) limits of particle systems and large scale front dynamics for turbulent combustion. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***