Nonlinear hamiltonian partial differential equations

在过去的十五年中，通过对多线性谐波分析和其他工具的系统应用，对基本非线性哈密顿偏微分方程（PDE）的洞察力出现了爆炸式的增长。对于具有粗糙初始数据的初值问题，现在已经有了一个鲁棒的实时局部理论。本研究计划旨在推动在最优正则性下的非线性哈密顿偏微分方程的最大时间理论的最新进展。特别是，拟议的研究在三个方面作出贡献：1。阐明局部时适定性的最优正则性。2. 在低正则性下建立全局实时适定性和长时间行为。3. 描述低正则性下的有限时间爆破行为。这些研究路线涉及等离子体物理、光学、流体动力学、量子力学等物理演化方程的数学共同特征，以及半导体、电信、航空航天等领域的工程应用。没有特定的科学或工程应用激发了所提出的研究，而是旨在为包括湍流，奇点形成，散射和递归在内的非线性现象的一般严格理论做出贡献。哈密顿PDE的广泛适用性，跨越当前科学和技术意义的不同领域，证明了我们的科学和工程基础设施所提出的研究的中心重要性。

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.