Aspects of Harmonic Analysis and Hamiltonian PDE's

主要研究者：Jean Bourgain，University of Illinois，U-CDMS-0322370摘要：PI建议研究哈密顿湍流中的问题，例如在薛定谔方程的光滑解中增长更高的Sobolev范数。 例如，考虑到具有周期性边界条件的二维散焦立方NLS，对于长时间的能量向更高模式的过渡，我们可以说些什么？目前看来，只有像上限这样的权力才是已知的。PI建议在此背景下探索动力系统方法。在线性薛定谔方程中，事情得到了更好的理解，部分原因是准周期局部化的进展。例如，PI最近建立了量子被踢转子的混沌扩散的情况下，小踢和几乎所有的参数值。他建议在这里进一步研究问题的大踢和估计本地化的长度。大多数偏微分方程研究的数学家起源于物理学或其他地方。它们被认为是对某些现象的模型，它们在这里的相关性经常被数字证实。虽然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，但它通常只是纯数学探索的开始。现在的目标是严格地研究这些方程作为数学对象，独立于任何先验假设，并试图恢复预期的行为作为数学定理。一方面，这种思路在过去导致了现代一些伟大的数学理论（可积性，湍流等）。但即便如此，在耗散和保守的政权中，仍然存在更多的挑战。在这个建议的重点在于扩散的哈密顿方程，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.