In this proposal we suggest to study open problems in regularity theory of the following partial differential equations: 1. strongly elliptic systems arising as Euler-Lagrange equations for multiple integrals in the Calculus of Variations; 2. three-dimensional Navier-Stokes equations and some related model equations; and 3. Complex Ginzburg-Landau Equations and Landau-Lifschitz equations. The regularity theory for these equations is mainly governed by two factors: the smoothing properties of the linearized operator, and the tendency of the non-linear part to produce singularities. In all the cases above there is a certain critical dimension below which (and, in many cases, at which) the natural energy estimates for the equations together with smoothing properties of the linearized operator are sufficient to guarantee regularity of solutions. For most of the equations above the critical dimension is two. The situation is more complicated above the critical dimension, when the specifics of each equation come much more into play, and not many results are known. The research will concentrate on problems in this relatively unmapped area.
The study of regularity theory for Partial Differential Equations (PDE) is usually motivated by the following factors: (1) A good regularity theory can serve as an important check that a PDE is appropriate for modeling a given phenomena. (2) The more information we have about regularity of solutions of a given PDE, the better chance we have to design a good numerical method for calculating its solutions on a computer. It is no coincidence that the equations for which we can calculate solutions accurately are more or less exactly those for which we have a good regularity theory. (Confirming that "There is nothing quite so practical as a good theory".) Roughly speaking, the more we know about solutions, the easier it is the avoid some of the many pitfalls into which numerical simulations can fall. The work suggested in this proposal will address basic open problems in regularity theory for important classes of equations with strong non-linearities. Many of these equations (such as the Navier-Stokes equations, or the Landau-Lifschitz equations) are of considerable practical interest.
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