Nonlinear Hamiltonian PDE

提出的研究旨在推动非线性哈密顿偏微分方程的最新进展，以实现三个目标：1。将局部时初值方法扩展到求解更复杂的pde。采用初值技术处理初边值问题。3. 构造非线性哈密顿偏微分方程的全局时间理论。该建议确定了一些具体问题，这些问题的解决有助于实现过去几十年的惊人进展所产生的三个目标。目标1的研究旨在通过执行一个增量研究计划，包括小x、b分母、空间各向异性和消失参数，扩展尖锐的一维微积分技术，以证明Bourgain的x、b空间中的多线性估计。最近，与Kenig合作开发了一种将初始边值问题重铸为具有边界强迫的初始值问题的技术。该方法的适用范围是目标2中提出的调查的主要主题。与Keel， Staffilani， Takaoka和Tao合作获得的KdV方程的L^2守恒定律的重新解释，通过使用多线性谐波分析和局部适位性机制构建几乎守恒的量，导致了一种显示全局适位性的新方法。提出的研究的第三个重点将利用这些量来理解非线性哈密顿偏微分方程的长期行为。拟议的研究并无具体的科学或工程用途；相反，其目的是为包括湍流、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。提出的研究旨在推动非线性哈密顿偏微分方程的最新进展，以实现三个目标：1。将局部时初值方法扩展到求解更复杂的pde。采用初值技术处理初边值问题。3. 构造非线性哈密顿偏微分方程的全局时间理论。该建议确定了一些具体问题，这些问题的解决有助于实现过去几十年的惊人进展所产生的三个目标。目标1的研究旨在通过执行一个增量研究计划，包括小x、b分母、空间各向异性和消失参数，扩展尖锐的一维微积分技术，以证明Bourgain的x、b空间中的多线性估计。最近，与Kenig合作开发了一种将初始边值问题重铸为具有边界强迫的初始值问题的技术。该方法的适用范围是目标2中提出的调查的主要主题。与Keel， Staffilani， Takaoka和Tao合作获得的KdV方程的L^2守恒定律的重新解释，通过使用多线性谐波分析和局部适位性机制构建几乎守恒的量，导致了一种显示全局适位性的新方法。提出的研究的第三个重点将利用这些量来理解非线性哈密顿偏微分方程的长期行为。拟议的研究并无具体的科学或工程用途；相反，其目的是为包括湍流、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。

The proposed research is designed to advance the recent progress on nonlinearHamiltonian PDE towards three goals:1. Extend the local-in-time initial value methods to solve more complicated PDE.2. Adapt the initial value techniques to treat initial-boundary value problems. 3. Construct a global-in-time theory of nonlinear Hamiltonian PDE.The proposal identifies specific problems whose solutions contribute to the three goals for which there are methods of attack emerging from the last decades' spectacular progress.The studies for Goal 1 aim to extend the sharp 1-dimensional calculus techniques for proving multilinear estimates in Bourgain's Xs,b spaces by carrying out an incremental research plan, involvingsmall Xs,b denominators, spatial anisotropy and vanishing parameters.A technique for recasting initial-boundary value problems as initialvalue problems with boundary forcing has recently been developed, incollaboration with Kenig. The range of applicability of this method isthe main topic of the proposed investigations toward Goal 2. A reinterpretation of the L^2 conservation law for the KdV equation, obtained in collaboration with Keel, Staffilani, Takaoka and Tao, has led to a new method for showing global wellposedness by constructing almost conserved quantities using multilinear harmonic analysis and the local wellposedness machinery. The third thrust of the proposed research will exploit these quantities to understand the long-time behavior of nonlinear Hamiltonian PDE.No specific scientific or engineering application motivates the proposed studies; rather the intention 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.The proposed research is designed to advance the recent progress on nonlinearHamiltonian PDE towards three goals:1. Extend the local-in-time initial value methods to solve more complicated PDE.2. Adapt the initial value techniques to treat initial-boundary value problems. 3. Construct a global-in-time theory of nonlinear Hamiltonian PDE.The proposal identifies specific problems whose solutions contribute to the three goals for which there are methods of attack emerging from the last decades' spectacular progress.The studies for Goal 1 aim to extend the sharp 1-dimensional calculus techniques for proving multilinear estimates in Bourgain's Xs,b spaces by carrying out an incremental research plan, involvingsmall Xs,b denominators, spatial anisotropy and vanishing parameters.A technique for recasting initial-boundary value problems as initialvalue problems with boundary forcing has recently been developed, incollaboration with Kenig. The range of applicability of this method isthe main topic of the proposed investigations toward Goal 2. A reinterpretation of the L^2 conservation law for the KdV equation, obtained in collaboration with Keel, Staffilani, Takaoka and Tao, has led to a new method for showing global wellposedness by constructing almost conserved quantities using multilinear harmonic analysis and the local wellposedness machinery. The third thrust of the proposed research will exploit these quantities to understand the long-time behavior of nonlinear Hamiltonian PDE.No specific scientific or engineering application motivates the proposed studies; rather the intention 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.