Nonlinear partial differential equations and applications

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This is a proposal to develop general methods to study nonlinear, hyperbolic and parabolic/elliptic partial differential equations. More precisely using techniques from analysis, partial differential equations and probability, the PI plans to continue his program working in the following four general areas: A. Fully nonlinear stochastic partial differential equations of first-and second- order . B. Phase Transitions. C. Turbulent reaction-diffusion equations and combustion. D. First- and second-order fully nonlinear, degenerate, elliptic and parabolic equations. Most of the partial differential equations considered in this proposal arise as models in continuum and statistical physics. These models appear in a variety of topics ranging from material sciences and phase transitions (motion of fronts, mesoscopic and macroscopic scales, homogenization), in fluid flows (turbulent reaction-diffusion and combustion), and in stochastic analysis (interacting particle systems, flows in random environments and with random velocities, and stochastic control). The qualitative analysis of these models contributes to the better understanding of the actual physical problems, and provides, in many cases, the foundation for the development of efficient numerical algorithms.