Wave Turbulence and Long-Time Dynamics of Dispersive Partial Differential Equations

伟大的美国物理学家理查德·费曼将湍流描述为“经典物理学中最重要的悬而未决的问题”。在这里，他指的是流体动力湍流，这是人们在日常生活中多次观察到的现象，特别是当人们在海洋上的船只或大气中的飞机上通过流体旅行时。尽管它是直观的表现，但科学上对湍流的理解还远远不能令人满意。一个相关的现象是“波动湍流”，它属于类似的问题，但对于涉及波浪相互作用的不同物理系统(例如，海洋或等离子体波)。这个项目的目的是从严格的数学观点更好地理解与波动湍流有关的某些现象，从而迈出将该理论建立在坚实的数学基础上的第一步。该项目研究了非线性色散和波动偏微分方程组的两种不同的长期行为。第一种状态可以用“非平衡动态”来描述，在这种状态下，解在平衡构型附近不会表现出长期的稳定性。这是在紧致域上提出的非线性色散方程的典型行为，其中色散效应不会转化为衰减，也是波浪湍流理论的自然背景。另一种长时间行为涉及欧几里德空间上提出的色散偏微分方程组，对于该方程，人们可以期望动力学有稳定的平衡点(定常解)。平凡平衡点(零解)的“渐近稳定性”到目前为止已被很好地理解，但非平凡平稳解的“渐近稳定性”还远未得到解决。该项目提出了几种研究途径，以提高我们对上述两个方向的理解。

The great American physicist Richard Feynman described turbulence as the "the most important unsolved problem of classical physics." Here, he was referring to hydrodynamic turbulence, which is the phenomenon one observes on numerous occasions in daily life, particularly when one travels through fluids either in watercraft on the ocean or in airplanes in the atmosphere. Despite its intuitive manifestations, the scientific understanding of turbulence is far from satisfactory. A related phenomenon is "wave turbulence," which pertains to similar problems but for different physical systems involving wave interactions (e.g.,ocean or plasma waves). The aim of this project is to gain a better understanding of certain phenomena pertaining to wave turbulence from a rigorous mathematical viewpoint and thereby to take the first steps towards putting the theory on solid mathematical foundations.The project addresses two different regimes of long-time behavior for nonlinear dispersive and wave partial differential equations. The first regime can be characterized by "out-of-equilibrium dynamics," in which solutions do not exhibit long-time stability around equilibrium configurations. This is the typical behavior of nonlinear dispersive equations posed on compact domains, where dispersive effects do not translate to decay, and is the natural setting of wave turbulence theory. The other regime of long-time behavior concerns dispersive partial differential equations posed on Euclidean space, for which one can hope to have stable equilibrium points for the dynamics (stationary solutions). The "asymptotic stability" of trivial equilibria (zero solutions) is mostly well understood by now, but that of nontrivial stationary solutions is far from settled. The project suggests several avenues of research to improve our understanding in the two aforementioned directions.