Nonlinear Hamiltonian PDE

本文的研究工作旨在推动非线性哈密顿偏微分方程的研究朝着以下三个方向发展：1.将时间局部初值法推广到求解更复杂的偏微分方程.采用初值方法处理初边值问题。3.建立一个非线性Hamilton偏微分方程的时间域整体理论。该方案确定了具体的问题，其解决方案有助于实现三个目标，在过去几十年的惊人进展中，出现了一些攻击方法。目标1的研究旨在通过实施一个增量研究计划，涉及小X，B算子、空间各向异性和消失参数。最近，与Kenig合作，发展了一种将初边值问题转化为有边界强迫的初边值问题的方法。这一方法的适用范围是为实现目标2而进行的拟议调查的主要议题。与基尔、斯塔菲拉尼、高冈和陶合作对KdV方程的L^2守恒律进行了重新解释，通过使用多线性调和分析和局部适定性机制构造几乎守恒量，得到了一种新的方法来证明整体适定性。研究的第三个重点是利用这些量来理解非线性哈密顿偏微分方程的长期行为，没有具体的科学或工程应用动机，而是旨在为非线性现象（包括湍流，奇异性形成，散射和递归）的一般严格理论做出贡献。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，表明了中央突出的拟议研究，我们的科学和工程基础设施。将时间局部初值法推广到求解更复杂的偏微分方程.采用初值方法处理初边值问题。3.建立一个非线性Hamilton偏微分方程的时间域整体理论。该方案确定了具体的问题，其解决方案有助于实现三个目标，在过去几十年的惊人进展中，出现了一些攻击方法。目标1的研究旨在通过实施一个增量研究计划，涉及小X，B算子、空间各向异性和消失参数。最近，与Kenig合作，发展了一种将初边值问题转化为有边界强迫的初边值问题的方法。这一方法的适用范围是为实现目标2而进行的拟议调查的主要议题。与基尔、斯塔菲拉尼、高冈和陶合作对KdV方程的L^2守恒律进行了重新解释，通过使用多线性调和分析和局部适定性机制构造几乎守恒量，得到了一种新的方法来证明整体适定性。研究的第三个重点是利用这些量来理解非线性哈密顿偏微分方程的长期行为，没有具体的科学或工程应用动机，而是旨在为非线性现象（包括湍流，奇异性形成，散射和递归）的一般严格理论做出贡献。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。

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.