Dispersive PDE at critical regularity

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目的主旨是进一步了解某些色散方程在临界正则性下的解的全局/大数据行为。更确切地说，主要研究者认为全球适定性和散射问题的非线性薛定谔，克莱因-戈登和（修改）Korteweg-de弗里斯方程的初始数据属于临界/低正则性Sobolev空间。非线性波动方程（NLW）和薛定谔方程（NLS）的临界正则性问题在过去的几年里引起了人们的广泛关注。这些工作已经开发了一套强大的工具和技术，旨在解决NLW和NLS在保守的临界正则性。该项目的主要目的是加强和扩大这一工具箱。近期目标包括治疗聚焦（低维）能量临界NLS和散焦/聚焦质量临界NLS，问题在于有点超出现有技术的范围（除了在径向数据的情况下）。第二，主要研究者希望测试迄今为止开发的工具箱对新困难的鲁棒性，例如临界正则性不对应于（强制性）守恒量的问题或对称性破坏的问题。 该项目的最后一部分是关注的全球适定性问题（修改）Korteweg-de弗里斯方程的初始数据在低正则性空间。由于完整的可积性技术，这个问题是更好地理解在周期的情况下比在非周期的。 首席研究员建议重新审视这些新的进展，由于Kappeler和Topalov从纯粹的偏微分方程的观点，希望发现一个适当的规范，将允许治疗的非周期性的情况下，在低regularity.The方程在这个项目中的调查有着丰富的历史。数学家和物理学家都对它们进行了研究，因为它们捕捉了某些物理行为的重要方面，同时保持了吸引人的简单性。 因此，它们是研究偏微分方程的新分析技术的温床。尽管要研究的方程相对于科学或工业的需要被大大简化，但主要研究者认为，对这些方程的研究将促进具有更广泛适用性的工具的发展，而即使是最微小的加速，也可以处理超临界方程（如著名的Navier-Stokes方程）的时代也是非常有益的。 在开发工具箱的同时，也在传播工具箱。 主要研究者将继续她在这方面的活动，包括维护一套关于这一材料的讲义。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The main thrust of this project is to further the understanding of the global/large data behavior of solutions to certain dispersive equations at critical regularity. More precisely, the principal investigator considers global well-posedness and scattering questions for nonlinear Schrodinger, Klein-Gordon, and (modified) Korteweg-de Vries equations for initial data belonging to critical/low-regularity Sobolev spaces. Critical-regularity problems for the nonlinear wave (NLW) and Schrodinger (NLS) equations have attracted considerable attention over the past few years. These works have developed a powerful set of tools and techniques meant to address NLW and NLS at conserved critical regularity. The main purpose of this project is to strengthen and broaden this toolbox. Immediate goals include treating the focusing (low-dimensional) energy-critical NLS and the defocusing/focusing mass-critical NLS, problems that lie a little beyond the reach of existing techniques (except in the case of radial data). Second, the principal investigator wishes to test the robustness of the toolbox developed thus far against new difficulties, such as problems for which the critical regularity does not correspond to a (coercive) conserved quantity or problems with broken symmetries. The last part of the project is concerned with the global well-posedness question for the (modified) Korteweg-de Vries equation for initial data in low regularity spaces. Thanks to complete integrability techniques, this problem is understood better in the periodic case than in the nonperiodic one. The principal investigator proposes to revisit these new advances due to Kappeler and Topalov from a purely partial differential equations point of view in the hope of discovering an appropriate gauge that would allow the treatment of the nonperiodic case at low regularity.The equations under investigation in this project have a rich history. They have been studied by mathematicians and physicists alike because they capture important facets of certain physical behaviors, while maintaining an attractive simplicity. As such, they serve as breeding grounds for new analytical techniques for studying partial differential equations. Although the equations to be investigated are drastically oversimplified relative to the needs of science or industry, the principal investigator believes that the study of these equations will foster the development of tools with much broader applicability, while even the tiniest hastening toward an era when supercritical equations such as the celebrated Navier-Stokes equation can be treated would be very beneficial indeed. Parallel to the development of a toolbox is its dissemination. The principal investigator will continue her activities in this direction, including the maintenance of a set of lecture notes on this material.