Nonlinear partial differential equations and applications

University Of Texas At Austin, Austin TX

Investigators

Abstract

PI: Takis Souganidis, University of Texas, Austin DMS-0244787 ABSTRACT The PI proposes to continue his program to develop methods to study nonlinear, parabolic/elliptic and hyperbolic, deterministic and stochastic partial differential equations (pde) arising as models in areas like continuum and statistical physics, biology, ecology, engineering, etc. The emphasis of this proposal is on the development of (rigorous) theories for (i) weak (stochastic viscosity) solutions of fully nonlinear, (degenerate) parabolic stochastic partial differential equations, and (ii) homogenization (averaging) of nonlinear, parabolic/elliptic and hyperbolic partial differential equations in spatio-temporal randomly heterogeneous media. The randomness is typically associated with singular dependence on the state variables and lack of compactness. Overcoming both issues requires the development of new methods and tools. Nonlinear stochastic pde are very often used to model complex phenomena, like turbulent combustion, nucleations, phase transitions at low temperature, etc., It is therefore self-evident that there is a pressing need for the development of a mathematical theory for such equations. The theory of stochastic viscosity solutions, which is been developed by the PI and Lions, is of fundamental importance, since it allows for the study of a completely new and, at the same time, very rich class of nonlinear stochastic pde arising in many of the aforementioned contexts. As this theory becomes more available, known and developed, it is expected that it will play an important role also in applications, since it will provide the necessary tools to analyze previously intractable models. The mathematical modeling of phenomena with many scales at the microscopic level, like turbulence, front propagation in phase transitions, spatial ecology, polymer growth, composite and perforated materials in mechanics, often necessitates the use of random pde with oscillating random parameters. At the macroscopic level these parameters "average out" and the phenomena are then described by an averaged (homogenized) equation. This leads naturally to questions of homogenization of nonlinear pde in spatio-temporal stationary ergodic settings, which, roughly speaking, are the most general settings where "self-averaging" is to be expected.