Nonlinear Partial Differential Equations with Applications

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9706231 Souganidis The PI proposes to study a broad range of problems, some motivated by applications in phase transitions, turbulent combustion, etc., which arise in the areas of hyperbolic and parabolic/elliptic nonlinear partial differential equations and probability. More precisely, using techniques from analysis, partial differential equations, and probability, the PI proposes to continue his program working in the following five general areas: A. Kinetic theory of conservation laws. B. Viscosity solutions of fully nonlinear elliptic and parabolic partial differential equations. C. Fully nonlinear stochastic partial differential equations. D. Phase transitions. E. Turbulent reaction-diffusion equations. Most of the questions to be studied in this project are of a theoretical nature, although much of the work in turbulent reaction-diffusion equations deals with modelling and computations. The questions related to this last topic as well as those related to phase transitions and fully nonlinear stochastic pde are important in material and environmental/oceanic sciences. Indeed these questions and the anticipated answers should provide appropriate mathematical models and insight for the study of basic questions in these sciences.