Nonlinear elliptic and parabolic equations with nonlocal effects

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This project concerns the analysis of elliptic and parabolic partial differential equations. The focus is on the regularity of solutions of equations whose diffusion is nonlocal, such as equations involving the fractional Laplacian. These equations arise, in particular, from models involving discontinuous stochastic processes. One important problem is to understand the equations arising from stochastic control problems and stochastic games involving discontinuous Levy processes. These equations are the fractional-order version of fully nonlinear elliptic and parabolic equations. The analysis of the regularity of integro-differential equations is not only interesting because it extends the theory of regularity of second-order partial differential equations, but also because it gives us a extra insight into what causes the regularization in elliptic and parabolic problems. Another mathematically interesting feature is that, since fractional diffusion has scaling properties different from those for second-order diffusion, the interplay between the first-order terms of the equation and the diffusion can become nontrivial at small scales. The principal investigator will study equations with advection and fractional diffusion, with an emphasis on the critical and supercritical regimes. Examples of this type of equations are the critical and supercritical quasi-geostrophic equation and the Burgers equation with fractional diffusion.The nonlocal equations presented in this project arise from models in engineering, finance and physics that involve long-range interactions. The most common situation in which this happens occurs in models involving discontinuous stochastic processes. For example, the price of a stock can suddenly jump from a value to a very different one. It has been suggested that it may be better to model stock prices with so-called discontinuous Levy processes than with diffusions. In that case, the equation involved in computing the value of an American option would be an obstacle problem for an integro-differential equation. Other nonlinear integro-differential equations arise from stochastic games, or from the physical phenomena known as anomalous diffusion. Nonlocal equations are also obtained from deterministic models. For example if one assumes that temperature is diffused quickly through the atmosphere, then this phenomenon creates a nonlocal diffusion effect on the surface of the earth.