Weighted Fourier analysis and dispersive equations
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Abstract
The goal of this thesis is to apply the theory of multilinear weighted Fourier estimates to nonlinear dispersive equations in order to tackle problems in regularity, well-posedness, and pointwise convergence of solutions. Dispersion of waves is a ubiquitous physical phenomenon that arises, among others, from problems in shallow-water propagation, nonlinear optics, quantum mechanics, and plasma physics. A natural tool for understanding the related physics is to study waves/signals simultaneously from both physical and spectral perspectives. Specifically, we will treat nonlinearities as multilinear operator perturbations, which (by the method of spacetime Fourier transforms), exhibit smoothing properties in norms defined to reflect the dispersive natures of the solutions. Our model equation is the quantum Zakharov system, which can be viewed as a variation on the cubic nonlinear Schrödinger equation (NLS). We investigate the model in various contexts (adiabatic limits, nonlinear Schrödinger limits, semi-classical limits). We additionally study a variation of Carleson's Fourier convergence problem in the context of pointwise convergence of the full Schrödinger operator with non-zero potential.