Research in Harmonic Analysis and Partial Differential Equations

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The principal investigator will undertake research in harmonic analysis and in the analysis of partial differential equations (PDE). Harmonic analysis has played major roles in the pure and applied sciences since Fourier's seminal work on the theory of heat diffusion, continuing on with Schrodinger's equation in quantum mechanics. It underlies a diverse array of tools widely used in science and engineering, and it offers the promise of further applications in the future. The proposed research deals with foundational issues that may help to underpin future applications. In PDE, the project will study the long-time dynamical properties of several fundamental equations describing diverse physical phenomenon. In particular, the nonlinear Schrodinger equation (NLS) models the transmission of data in fiber optic communication systems, and the Korteweg-de Vries equation (KdV) models surface water waves as well as ion-acoustic waves in a cold plasma. The so-called fractional NLS is used as a model describing charge transport in bio polymers like DNA. Proposed problems on near-linear behavior and smoothing are directly motivated by real world engineering problems in fiber optic communication systems, and the methods used are likely to be useful in a range of applications.In harmonic analysis this project focuses on problems in Euclidean spaces centered around Lebesgue norm inequalities. One subject of on-going research is the Fourier restriction phenomenon and its applications to problems in PDE and geometric measure theory. In the PDE component of the project, the focus is on the dynamical properties of linear and nonlinear dispersive equations. Subjects of interest here are dispersive decay and smoothing estimates for Schrodinger and wave equations, and their applications to the stability problem for their nonlinear counterparts. Another topic is the regularity properties of the solutions of nonlinear dispersive PDE such as the KdV equation, the Zakharov system, and the fractional NLS. The principal investigator will continue to explore the smoothing effect of the dispersive linear group on bounded domains, and he will study applications to the regularity properties and long-time dynamics of the nonlinear solutions. Proposed applications are on the existence and regularity of global attractors, dispersive quantization/Talbot effect, bounds for higher order Sobolev norms, and controllability.