Aspects of regularity theory for PDE

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The project addresses several open problems in the theory of partial differential equations. The problems arise primarily in the context of equations used in fluid mechanics (the Navier-Stokes equations and Euler's equations) or in the calculus of variations (regularity of minimizers of multi-dimensional variational integrals for vector-valued functions). For the Navier-Stokes equations the study will include the following areas of emphasis: (1) long-distance behavior of steady-state solutions; (2) regularity for special classes of solutions, such as the axi-symmetric solutions; and (3) regularity of related linear equations with low-regularity coefficients. In the case of Euler's equations, the project will focus on the following topics: (1) existence of periodic and quasi-periodic solutions; (2) the structure of the set of the steady-states of the two-dimensional equations; and (3) the relevance of various steady-states of the two-dimensional equations for two-dimensional statistical theories. The problems in the calculus of variations concern the stability of singularities. Unlike in the scalar case, the minimizers of regular variational functionals for vector-valued functions can have singularities. How stable are these singularities? This question will be addressed. The topic also has connections to nonlinear elasticity. Theoretical research in partial differential equations ultimately has a very practical goal, which is the understanding and prediction of behavior of solutions of the equations. For example, fluid flows are usually described by the Navier-Stokes equations. Do the solutions of these equations correctly describe what we see "in practice," and can we use the equations to make precise predictions about the flows? From what we know about these equations, the answers seem to be yes, but the mathematics of the equations is still not very well understood. Indeed, we do not yet have satisfactory mathematical explanations for the behavior of solutions that can be observed either in experiments or in computer simulations. The information we need to obtain from the computations can usually be formulated in simple and concrete terms. For instance, at what speed will an aircraft stall? The question is simple, whereas the behavior of the solutions underlying this question is complicated. Can we somehow manage the complexity by finding the most important mathematical parameters of the flow that are still "controlable"? When we study regularity of solutions, the situation is quite similar: there seems to be a very large spectrum of behaviors that solutions can exhibit. Can we obtain some control of the solutions by focusing on relatively few "right" parameters? What are the right parameters? This research project focuses on questions of this nature. If we can identify some good quantities that govern the behavior of the solutions and that we are able to control, then it becomes much easier to calculate the solutions themselves, since the theoretical information tells us where to focus our computational resources.