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。该项目的主要目的是进一步理解某些色散方程在临界规则下解的全局/大数据行为。更准确地说，主要研究者考虑非线性薛定谔、Klein-Gordon和（修正的）Korteweg-de Vries方程的全局适定性和散射问题，这些方程属于临界/低正则Sobolev空间。非线性波（NLW）和薛定谔（NLS）方程的临界正则性问题在过去的几年中引起了相当大的关注。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在保守的临界规律。这个项目的主要目的是加强和扩大这个工具箱。当前的目标包括处理聚焦（低维）能量临界NLS和散焦/聚焦质量临界NLS，这些问题超出了现有技术的范围（除了径向数据的情况）。其次，首席研究员希望测试迄今为止开发的工具箱的鲁棒性，以应对新的困难，例如临界正则性不对应于（强制）守恒量的问题或对称性破碎的问题。项目的最后一部分涉及低正则性空间初始数据的（修正）Korteweg-de Vries方程的全局适定性问题。由于完全可积性技术，这个问题在周期情况下比在非周期情况下更容易理解。首席研究员建议从纯粹的偏微分方程的角度重新审视Kappeler和Topalov的这些新进展，以期发现一种适当的标准，允许在低正则性下处理非周期性情况。在这个项目中所研究的方程有着丰富的历史。数学家和物理学家都对它们进行了研究，因为它们捕捉到了某些物理行为的重要方面，同时又保持了一种吸引人的简单性。因此，它们为研究偏微分方程的新分析技术提供了温床。尽管要研究的方程相对于科学或工业的需要来说过于简化了，但首席研究员相信，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，也会使像著名的纳维-斯托克斯方程这样的超临界方程能够得到处理，这将是非常有益的。与工具箱的发展并行的是它的传播。首席研究员将继续她在这个方向上的活动，包括维护一套关于这些材料的课堂讲稿。

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.