This mathematics research project is in the general area of partial differential equations. The focus is on the study of the regularity properties of elliptic and parabolic partial differential equations, and their connections with the well posedness of such problems; parts of the project concern non local diffusions. The partial differential equations investigated in this project arise from stochastic models dealing with discontinuous processes. A number of models from finance, physics, population dynamics, chemistry and biology involve non local diffusions. Another related direction of research concerns the interaction between local or non local diffusion with advection. There are a number of advection-diffusion models from physics that are not currently well understood mathematically. Typical examples of such phenomena arise in the study of equations that model fluids dynamics. The principal investigator Luis Silvestre will study estimates for advection-diffusion equations geared towards an improved understanding of the active scalar equations arising from models in fluid dynamics. Silvestre will also investigate a number of questions related to fully nonlinear elliptic partial differential equations that occur in the study of zero-sum stochastic games.
This mathematics research projects studies the behavior of so-called non-local equations. Such equations arise in any physical model with long range interactions. They also arise naturally as the equations governing any probabilistic model whose values may take long jumps. A common example is the modeling of stock prices in financial mathematics, which could sporadically take sudden changes. Non local equations appear in myriad of models from physics, finance, social sciences and biology. Their applications may range from the valuation of financial options, to the effective computation of protein docking, which is useful in the design of medicinal drugs, and even in the modeling of the flight of birds. The development of a general theory of non local partial differential equations has seen great progress in recent years, which goes side to side with an increasing number of applications. The equations describing the dynamics of fluids present very difficult mathematical questions. Luis Silvestre (the principal investigator in this project) will also study certain diffusion equations, local and non local, in moving fluids. For the educational part of this proposal, Silvestre will organize summer schools, design an improved partial differential equations class for undergraduates, as well as teaching a graduate class; Silvestre will also organize conferences, and will mentor graduate students and postdoctoral fellows.
