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（Richard Feynman）将湍流描述为“经典物理学中最重要的未解决问题”。在这里，他指的是流体动力学湍流，这是人们在日常生活中经常观察到的现象，特别是当人们在海洋上的船只或在大气中的飞机上穿越流体时。尽管有直观的表现，但对湍流的科学理解远不能令人满意。一个相关的现象是“波浪湍流”，它适用于类似的问题，但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）。该项目的目的是从严格的数学角度更好地理解与波浪湍流有关的某些现象，从而迈出将理论建立在坚实的数学基础上的第一步。该项目解决了非线性色散和波动偏微分方程的两种不同的长期行为。第一种状态可以用“非平衡动力学”来描述，在这种状态下，溶液在平衡构型周围不表现出长期的稳定性。这是非线性色散方程在紧致域上的典型行为，色散效应不会转化为衰减，并且是波浪湍流理论的自然设置。长期行为的另一种形式涉及欧几里得空间上的色散偏微分方程，对于这种方程，人们可以希望有稳定的动力学平衡点（平稳解）。平凡平衡点（零解）的“渐近稳定性”现在已经得到了很好的理解，但非平凡平稳解的“渐近稳定性”还远未解决。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.