Aspects of Harmonic Analysis and Hamiltonian PDE's

PI：Jean Bourain，伊利诺伊大学，U-CDMS-0322370摘要：PI建议研究哈密顿湍流中的问题，如薛定谔方程光滑解的高Sobolv范数的增长。例如，考虑具有周期边界条件的二维散焦立方NLS，关于能量在大时间内向更高模式的跃迁，人们能说什么呢？目前似乎只有像上限这样的权力是已知的。在这种情况下，PI建议探索动力系统方法。在线性薛定谔方程中，事物被更好地理解，部分原因是准周期局部化的进展。例如，PI最近证实，对于小踢和几乎所有的参数值，量子踢转子不存在混沌扩散。他建议在这里进一步研究大踢点和估计局部化长度的问题。数学家研究的大多数偏微分方程式都起源于物理学或其他地方。它们被认为是某些现象的模型，而它们在这里的相关性通常是通过数字来证实的。但是，尽管从现象学到数学建模的这一发展阶段被大多数科学家认为是令人满意的，但这通常只是纯粹数学探索的开始。现在的目标是将这些方程作为数学对象严格研究，而不依赖于任何先验假设，并试图将预期行为恢复为数学定理。一方面，这一思路在过去导致了现代的一些伟大的数学理论(可积性、湍流等)。但即便如此，更多的挑战仍然存在，在耗散和保守的政权中也是如此。这一建议的重点在于哈密顿方程中的扩散，特别是薛定谔方程。

PI: Jean Bourgain, University of Illinois, U-CDMS-0322370Abstract: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.