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.
Partial differential equations are used to model a variety of phenomena in nature, social sciences, finance and biology, among others. A type of equations whose study has grown extensively in the last fifteen years are called "non local equations". These equations are used in models affected by long range interactions or by random processes which make sudden changes. For example, in financial mathematics they are used in pricing models for options in a market which could take sudden jumps. There are several other models in the physical sciences which are modelled using non local equations but the reasons are more subtle: gas dynamics, non local electrostatic models for protein docking (these models are being studied for their potential use in pharmaceutical drug development), diffusion through the athmosphere, image processing, etc. The main focus of this project was on the study of regularity properties of non local equations. A large class of partial differential equations have regularization effects, which means that the values of the solution become regular or smooth as time evolves. Think for example that if we put together several objects of different temperature in a closed box, after some time the temperatures will tend to even up, and this effect is stronger for the objects that are next to each other. There are several ways to quantify this regularization effect for solutions of partial differential equations. Of course, it strongly depends of the equation under study. The issue of regularity is also closely related to other mathematical questions for differential equations like the existence, uniqueness and stability of solutions. It is also of vital importance in order to determine the accuracy of a numerical approximation obtained in a computer. In this project we obtained regularity results in a variety of non local equations modeling problems in stochastic control and fluid mechanics among others. We also obtained some results for more classical "local" equations, which are non linear and also have regularization effects.
