This project focuses on several topics in partial differential equations, both single equations and systems of equations, and considers problems involving fully nonlinear elliptic equations, evolutionary equations that arise in fluid mechanics, and error bounds for numerical approximations to solutions of fully nonlinear elliptic and parabolic equations. The first part of the project deals with the regularity of a family of fully nonlinear degenerate elliptic equations. These equations turn up in the optimal mass transport problem and in geometry. The second part is the investigation of global regularity, blow-up phenomena, partial regularity, and regularity criteria for several models of nonlinear parabolic equations in fluid mechanics. These include the well-known Navier-Stokes equations, the quasi-geostrophic equations, and some other related models. The third component is a study of error estimates for finite-difference approximations to solutions of fully nonlinear and possibly degenerate elliptic and parabolic Bellman equations. Bellman equations surface in many areas of mathematics (e.g., control theory, mathematical finance, differential geometry). It is thus a natural problem to seek numerical methods for approximating solutions to such equations. The fourth portion of the project concentrates on the theory of second-order linear elliptic and parabolic systems. A crucial difference between scalar equations and systems is that the classical maximum principle and the Harnack inequality are no longer applicable for systems.
The projects described in the previous paragraph are interesting not only from a mathematical perspective. They are also of central importance and have significant applications in areas such as physics, economics, and finance. For instance, one application occurs in the area of fluid mechanics and turbulence, where one would like to estimate external flow over all kind of vehicles such as cars, airplanes, ships and submarines, to understand the formation of hurricanes, or to predict the earth's atmospheric circulation. These examples are all related to the three-dimensional Navier-Stokes equations, the theory of which is far from complete. Another important application is in finance, say, to calculate numerically the expected performance of a stock portfolio. For example, the Black-Scholes option pricing model in finance, which is a well-known mathematical model of the market for equity, is closely related to the partial differential equation that governs heart flow. Moreover, many of these problems are naturally modeled as coupled systems of partial differential systems rather than as single equations. The results obtained under this research will help to improve the mathematical models that are used in such applications.