Aspects of Harmonic Analysis and Hamiltonian PDE's

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

PI: Jean Bourgain, University of Illinois, U-C DMS-0322370 Abstract: The PI proposes to study issues in Hamiltonian turbulence such as growth of higher Sobolev norms in smooth solutions of Schroedinger equations. Considering for instance the 2D defocusing cubic NLS with periodic boundary conditions, what can one say about transition of energy to higher modes for large time? Only power like upper bounds seem presently known. The PI proposes to explore dynamical systems methods in this context. In linear Schroedinger equations, things are better understood, partly due to progress in quasi-periodic localization. For instance the PI established recently the absence of chaotic diffusion for the quantum kicked rotor for small kicks and almost all values of the parameters. He proposes here to study further the problem of large kicks and estimating localization lengths. Most partial differential equations studied by mathematicians originate from Physics or elsewhere. They are supposed to model certain phenomena and their relevance here is often confirmed numerically. But while this stage of development from phenomenology to mathematical modeling is by most scientists considered satisfactory, it is usually only the beginning of purely mathematical exploration. The aim now is to study these equations rigorously as mathematical objects, independently of any a priori assumptions, and to try to recover the expected behaviour as mathematical theorems. On one hand, this line of thought has in the past led to some of the great mathematical theories of modern days (integrability, turbulence, etc.). But even so, much more challenges remain, as well in the dissipative as conservative regime. The emphasis in this proposal lies on diffusion in Hamiltonian equations, in particular the Schroedinger equation.