Nonlinear Partial Differential Equations and Applications

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SouganidisThe 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 common to have only \statistical? (random) and not \exact? (deterministic) information. In addition, incorporating the fluctuations of several physical quantities leads to equations with \singular? (white noise type) dependence on some of the variables. In this context, random homogenization and stochastic pde become the natural mathematical objects. 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. Viscosity solutions of nonlinear ¯rst- and second-order pde have become a classical tool for the study of many applications. It is therefore important to improve the understanding of their qualitative properties. In biology, recent experiments at the molecular scale 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 behavior. The PI proposes to continue his program to develop methods to study nonlinear deterministic and stochastic pde arising in continuum and statistical physics, biology, engineering, etc.. The emphasis of the proposal is on (i) the development of theories for weak (stochastic viscosity) solutions of fully nonlinear, (degenerate) parabolic stochastic pde, and the homogenization of nonlinear, parabolic/elliptic and hyperbolic pde in spatiotemporal random media, (ii) the study of problems related to viscosity solutions (rates of convergence, averaging, large deviations in infinite dimensions, etc.), and (iii) the analysis of models for motor and concentration effects in mathematical biology. First- and second-order, stochastic pde and stochastic homogenization arise in models for a wide variety of phenomena and applications including turbulence, phase transitions and front propagation in random media with/out random velocities, 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 pde. As the theory 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 resurgence of interest in homogenization in random media. The novel tools proposed to be developed are expected to become the standard methodology in the field. Providing a unified (analytic) treatment, based on viscosity solutions, to a class of large deviation problems in infinite dimensions will lead to a better understanding of an important area with applications to particle systems, random matrices, phase transitions, etc.. In mathematical biology, the proposed work is expected to enhance the understanding of concrete phenomena like bio-motor behavior and concentration effects.