Analytical and geometrical problems involving non linear diffusion processes

University Of Texas At Austin, Austin TX

Investigators

Abstract

The purpose of this research project is to develop a mathematical understanding of a number of scientific phenomena that are modeled by nonlinear integro-differential equations. The main thrust of the project concentrates on solutions of nonlinear problems involving anomalous (in particular integral) diffusion processes arising, for instance, in phase transitions, fluid dynamics, game theory and optimal control. Specific examples of the type of phenomena being studied include, in the continuum mechanics side, the analysis of equations modeling surface phenomena, like ocean-atmosphere interaction (the quasigeostrophic equation) or planar dislocation propagation, and diffusion at multiple scales, like polymers or turbulent flow, while on the probability side, problems of optimal control in game theory and finance, involving random variables that jump discontinuously (Levy processes or Levy walks) in changing environments. The issues to be investigated in this research project have a certain universality in the sense that the same paradigm reappears from geometry and analysis, to fluid dynamics and material sciences, to financial mathematics and more recently biology and stochastic geometry. The proposed research program will improve the interplay between modeling and phenomena in these areas. Understanding their basic structure will promote interdisciplinary research.