Nonlinear hamiltonian partial differential equations

在过去的15年里，通过系统地应用多线性调和分析和其他工具，对基本非线性哈密顿偏微分方程（PDE）的深入了解发生了爆炸。一个强大的本地时间理论的初值问题与粗糙的初始数据现在到位。这项研究的目的是推进最大时间理论的最佳正则性下的非线性哈密顿偏微分方程的最新进展。特别是，拟议的研究贡献沿着三条线：1。阐明局部时间适定性的最优正则性。2.在低规律性下建立全局时间适定性和长时间行为。3.描述在低正则性下的有限时间爆破行为。这些调查线解决了等离子体物理学，光学，流体动力学，量子力学等中物理演化方程的数学共同特征，半导体、电信、航空航天等领域的工程应用。没有具体的科学或工程应用激励拟议的研究，而不是目的是对非线性现象，包括湍流，奇点形成，散射和复发的一般严格的理论作出贡献。哈密顿偏微分方程的广泛适用性，在当前的科学和技术意义的不同领域，表明我们的科学和工程基础设施的拟议研究的中心突出。

An explosion of insight into basic nonlinear Hamiltonian partial differential equations (PDE) occurred in the last fifteen years through a systematic application of multilinear harmonic analysis and other tools. A robust local-in-time theory for initial value problems with rough initial data is now in place. This research proposal is designed to advance the recent progress towards a maximal-in-time theory for nonlinear Hamiltonian PDE under optimal regularity properties. In particular, the proposed research contributes along three lines: 1. Clarify optimal regularity for local-in-time well-posedness. 2. Establish global-in-time well-posedness and long time behavior at low regularity. 3. Describe finite time blowup behavior at low regularity. These lines of investigation addresses mathematically common features of physical evolution equations in plasma physics, optics, fluid dynamics, quantum mechanics, etc., with engineering applications in semiconductors, telecommunications, aerospace, and beyond. No specific scientific or engineering application motivates the proposed studies rather the aim is to contribute toward a general rigorous theory of nonlinear phenomena including turbulence, singularity formation, scattering and recurrence. The widespread applicability of Hamiltonian PDE, across diverse fields of current scientific and technological significance, demonstrates the central prominence of the proposed research to our science and engineering infrastructure.