Aspects of Harmonic Analysis and Hamiltonian PDE's

摘要：PI提出研究hamilton湍流中薛定谔方程光滑解中高Sobolev范数的增长等问题。例如，考虑到具有周期性边界条件的二维离焦立方NLS，对于大时间内能量向更高模态的跃迁，我们能说些什么呢？目前看来，只有上界这样的力量是已知的。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域化的进展。例如，PI最近建立了量子踢转子对于小踢和几乎所有参数值的混沌扩散的不存在。在此，他建议进一步研究大踢脚问题和定位长度的估计。数学家研究的大多数偏微分方程起源于物理学或其他领域。它们应该模拟某些现象，它们在这里的相关性通常在数字上得到证实。但是，虽然大多数科学家认为从现象学到数学建模的这一发展阶段是令人满意的，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地研究，独立于任何先验假设，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，更多的挑战依然存在，不论是在耗散的还是保守的政权中。这个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.