Analysis of evolution equations and related problems

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The main goal of this project is to study the long-time behavior of certain linear and nonlinear evolution equations. In the linear case, the problems of interest include the following: the long-time behavior and growth of Sobolev norms of solutions to evolution equations when the value of the coupling constant is generic; applications to multidimensional scattering for Schrodinger operators with slowly decaying or random potentials; the analysis of the spatial asymptotics of Green's function for Schrodinger equations; and sharp results for the dynamics in one-dimensional wave equations with decaying potentials. The nonlinear evolution equations to be studied in this project have their origins in two-dimensional incompressible fluid dynamics. Specifically, the equations to be studied are the Euler equation and the surface quasigeostrophic equation. The project will address the issue of instability for these equations. In the case of the Euler equation, the problem of optimal growth of the Sobolev norms of vorticity will be studied, while in the quasigeostrophic setting the scenario of blow-up in finite time will be considered. To make a progress, the principal investigator will use the tools of harmonic analysis (multilinear operators, potential theory, harmonic measure, singular integrals), approximation theory (polynomials orthogonal on the circle and on the real line), probability (Ito's calculus), and spectral theory for self-adjoint operators and hyperbolic pencils. This project will focus on mathematical problems that are central to quantum and fluid mechanics. Quantum mechanics, which was created and developed in the last century, is a basic branch of the modern physics, and the dynamics of fluids is another branch of physics studied as early as the eighteenth century by Leonhard Euler. The analysis of evolution equations and wave propagation in the presence of rough or random medium suggested in this project is a central problem of quantum mechanics, so this research has a potential impact on the development of that field. One of the most intriguing problems in fluid dynamics is the problem of singularity formation. This phenomenon is ubiquitous in nature and one goal of this project is to study its mechanism mathematically by focusing on some simplified two-dimensional models. To accomplish these goals, tools from various areas of mathematics will be refined and applied, which will advance these fields as well. The work on the project will include mentoring graduate students and coaching undergraduate research teams. This will have an additional impact on human resource development.