Nonlinear Partial Differential Equations and Applications

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Nonlinear Partial Differential Equations and Applications Abstract of Proposed ResearchPanagiotis E Souganidis One of the most challenging problems in applied sciences is the modeling of phenomena with many degrees of freedom (scales), which are expected to have some averaged (macroscopic), perhaps random, behavior. The multi-scales and complexity of the problems in nature often necessitate the use of random media. In many applications, it is also common to have only ``statistical'' (random) and not ``exact'' (deterministic) information. In addition, the modeling of the fluctuations of the physically relative quantities leads to equations with ``singular'' (white noise type) dependence on some of the variables. In this context, random homogenization and stochastic partial differential equations become the natural mathematical objects. The randomness is associated with singular dependence on the state variables and lack of compactness, two facts that give rise to challenging mathematical problems. Overcoming them requires the development of new methods and techniques. 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, which affect the experimentally observed behavior.The PI proposes to continue his program to develop methods to study nonlinear deterministic (parabolic/elliptic and hyperbolic) deterministic and stochastic partial differential equations arising in models in areas such as continuum and statistical physics, biology, engineering, etc. The emphasis of the proposal is on the development of theories for (i) weak (stochastic viscosity) solutions of fully nonlinear, (degenerate) parabolicstochastic pde, (ii) the homogenization of nonlinear, parabolic/elliptic and hyperbolic pde in spatio-temporal random media, (iii) the study of properties (regularity, error estimates) of viscosity solutions, and (iv) the analysis of some models in mathematical biology.