Nonlinear hamiltonian partial differential equations

在过去的十五年里，通过系统地应用多线性调和分析和其他工具，人们对基本的非线性哈密尔顿偏微分方程(PDE)的洞察力出现了爆炸性的增长。对于具有粗糙初始数据的初值问题，一个稳健的局部时间理论现在已经到位。这一研究方案旨在推进最优正则性下非线性哈密顿偏微分方程解的最大时间理论的最新进展。具体地说，本研究的贡献有三个方面：1.阐明了局部时间适定性的最优正则性。2.建立低规则性的全局适定性和长时间行为。3.描述了低正则性下的有限时间爆破行为。这些研究方向解决了等离子体物理、光学、流体动力学、量子力学等领域中物理演化方程的数学共同特征，以及在半导体、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来推动拟议的研究，其目的是为包括湍流、奇点形成、散射和重现在内的非线性现象的一般严格理论做出贡献。哈密顿偏微分方程在当前科学和技术意义的不同领域的广泛适用性，表明了拟议的研究对我们的科学和工程基础设施的核心重要性。

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.