Nonlinear Partial Differential Equations and Applications

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The modeling of multi-scale phenomena necessitates the use of random media (periodicity is a rather restrictive structure for many applications) and requires the study of averaged (mesoscopic and macroscopic) behaviors. For complex phenomena, it is also very often the case that most of the available information is ``statistical'' (random) and not ``exact'' (deterministic). Furthermore incorporating the fluctuations of several physical quantities leads to equationsmwith ``singular'' (white noise type) and ``random'' dependence on some of the variables. In this context, random homogenization and stochastic partial differential equation become the natural mathematical objects. From the mathematical point of view, the randomness is associated with singular dependence on the state variables and lack of compactness both giving rise to challenging mathematical problems. Overcoming them requires the development of new methods and techniques. In biology, experiments at the molecular scale as well as new theories have led to new sophisticated mathematical models. Novel tools and ideas are needed to study these problems further and to identify all the relevant regimes/scales of the parameters affecting the experimentally observed and theoretically conjectured behavior.The PI proposes to continue his program to develop methods to study nonlinear deterministic and stochastic partial differential equation arising in continuum and statistical physics, biology, engineering, etc.. The emphasis is on the development of theories for (i) the homogenization of nonlinear, parabolic/elliptic and hyperbolic partial differential equation in spatio-temporal random media and applications to mean field games and front propagation and (ii) weak (stochastic viscosity) solutions of fully nonlinear, (degenerate) parabolic stochastic partial differential equation, and (iii) the analysis of models for adaptive dynamics in mathematical biology.The development of mathematical tools to study complex phenomena in multi-scale environments, especially when very often the only available information/data are statistical (random), is of the out most importance. Nonlinear, first- and second-order, stochastic partial differential equation and stochastic homogenization arise in models for a wide variety of phenomena and applications including mean field games, turbulence, phase transitions and front propagation in random media, nucleations in physics, macroscopic limits of particle systems, stochastic control theory, stochastic control with partial observations, financial mathematics, etc.. The theory of stochastic viscosity solutions is important. It allows for the study of a completely new class of fully nonlinear stochastic partial differential equation. As the subject develops further, it is expected that it will play a crucial role in applied areas by providing the necessary tools to analyze previously intractable models. There has been a resurgence in interest in homogenization in random media. The novel tools and methods that have already been and are proposed to be developed are expected to become the standardmethodology in the field. In mathematical biology, the proposed work is expected to enhance the understanding of concrete phenomena in adaptive dynamics. All the proposed areas of work are current, important and very active. The PI, who currently has four graduate students (two female) and two postdocs (one female) plans to continue his educational and training activities aiming towards the development of high quality researchers working in problems in the proposed areas as well as nonlinear partial differential equation in general.