Problems in Nonlinear Partial Differential Equations

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Broadly speaking, the project addresses open problems in regularity theory of elliptic and parabolic nonlinear systems of partial differential equations, as well as several problems from the theory of incompressible Euler equations. Most of the elliptic and parabolic problems are formulated in the context of the incompressible Navier-Stokes equations, but various model equations will also be studied and the insight obtained from the model equations is expected to be important. The main emphasis is on developing methods and techniques for problems that are out of the reach of perturbation theory. For certain classes of equations (such as, for example, the Navier-Stokes equation) such problems are of considerable theoretical and practical importance. Partial differential equations represent a basic tools for modeling many natural and technological phenomena. For instance, the equations describing fluid flow (on which an important part of this project will concentrate) are used in weather prediction, climate research, and various technological applications, such as the design of optimal aerodynamical shapes. The equations of fluid dynamics are notoriously difficult to solve, even with the help of the largest computers. Some of these difficulties are intrinsic, but some are due to the fact that our mathematical understanding of the equations themselves is incomplete. Advances in theoretical understanding of partial differential equations can lead to substantial improvement in practical methods for solving them. In the final analysis, the main task of the theoretical investigation is to find some simple and natural parameters that control the behavior of the solutions. Once a good set of such parameters is known, it becomes much easier to design practical methods for calculating the solutions. Reason: armed with such parameters a researcher knows what is important and what is not when one tracks a solution. The researcher can then focus computational power on the aspects that really matter. The proposed research can have a significant impact in this direction.