Harmonic Analysis, Mathematical Physics, and Nonlinear PDE

点击翻译按钮获取中文摘要

Harmonic Analysis, Mathematical Physics and Nonlinear PDEAbstract of Proposed ResearchWilhelm Schlag This project is explore the long time behavior, or prove that singularities will form in finite time, of solutions of equations that arise in mathematical physics. The equations under consideration typically admit nonlinear bound states (solitons or instantons) and much research has recently been devoted to the perturbative analysis of such solutions. It is well known that solutions of nonlinear Schroedinger and wave equations of the focusing type may blow up in finite time (if the energy of the data is negative, for example). It turns out, however, that global solutions exist if the data belong to a submanifold of finite co-dimension (a "center-stable" manifold in the language of dynamical systems). We shall investigate whether there is a manifold that divides a region of blow-up from one of scattering. Our goal is to obtain a deeper understanding of blow-up phenomena. Recently, progress was made for the critical wave-map equation into the two-dimensional sphere with regard to blow-up. It can be shown in a very precise and quantitative way that blow-up for this equation occurs through the bubbling off of energy via a non-constant harmonic map. Moreover, it turns out that the blow-up rate can be prescribed a priori. Similar phenomena occur for the semi-linear energy critical focusing equation in three plus one dimensions. Currently we do not understand which classes of equations admit this kind of phenomenon.Much of the success of science and engineering lies with its effective use of mathematical tools, both in terms of modeling and for computational simulation. The nonlinear Schroedinger equation arises in various applications in optics where a bound state (soliton) for represents a particle, or beam, that travels for a long time without disintegrating. An important issue is to understand the stability or instability of these solitons. That is, whether they persist under small perturbations or not? The theoretical understanding of these issues is very difficult, and is requiring new insights into mathematical problems. This project will investigate these problems and develop methods that may be used by practicing scientists and engineers.