Hard Problems in Hard Analysis

Washington University, Saint Louis MO

Investigators

Abstract

We investigate four major unsolved problems in or bordering on harmonic analysis. These are the Kakeya problem, the Lipschitz differentiation problem, the restriction problem, and global solvability for the Navier Stokes equation. For the Kakeya problem, we continue our work with Tao on improving exponents in the sums-differences approach. For the problem of differentiation by Lipschitz vector fields, we try to apply our work on maximal functions in arbitrary directions to understand what are the limitations on a counterexample and hopefully that none can exist. For the restriction problem, we try to apply the new results on Kakeya and to better understand Bourgain's machine for converting Kakeya results to ones about restriction. For Navier Stokes, we first discretize everything in the form of a kind of generalized wavelet coefficients. In work with Pavlovic, this has already produced a generalization of the Caffarelli-Kohn-Nirenberg theorem to the case of hyperdissipation. We hope from this point of view to discover a sort of local dispersion property for the cascading effect from the nonlinear term. Then we hope to tie in the Clay problem with a dyadic model in which this dispersion is a given. As might be imagined, we are unlikely to solve all these problems. Analysis concerns the proof of estimates on interesting systems by examing the contributions of all their parts. One such system is the Navier Stokes equation which governs the behaviour of incompressible viscous fluids. An important open problem is whether this equation starting with smooth initial data can develop singularities without a forcing term. This would akin to a cyclone beginning spontaneously in one's bathtub. It seems rather unlikely but the tools of analysis are not yet strong enough to rule it out. Our approach is to discretize the problem, that is to try to approximate the problem by one about a finite number of objects and investigate possible interactions of those objects by means of combinatorics. Most physically arising mathematics can be looked at this way because matter is not continuous but rather composed of particles. We will work on the above problem and some other important problems which may be approached with the same point of view.