Dispersive PDE at critical regularity

临界正则性的色散偏微分方程

• 批准号: 0901166
• 负责人: Monica Visan
• 金额: $ 15.4万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-15 至 2009-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901166&HistoricalAwards=false
• 关键词: Dispersive; PDE; critical; regularity

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.

该奖项根据 2009 年美国复苏和再投资法案（公法 111-5）提供资金。该项目的主要目的是进一步了解某些色散方程在临界规律下的解的全局/大数据行为。更准确地说，主要研究者考虑了属于临界/低正则性 Sobolev 空间的初始数据的非线性薛定谔、克莱因-戈登和（修改的）Korteweg-de Vries 方程的全局适定性和散射问题。非线性波（NLW）和薛定谔（NLS）方程的临界正则性问题在过去几年引起了广泛的关注。这些工作开发了一套强大的工具和技术，旨在以保守的关键规律解决 NLW 和 NLS 问题。该项目的主要目的是加强和扩大该工具箱。近期目标包括处理聚焦（低维）能量关键 NLS 和散焦/聚焦质量关键 NLS，这些问题超出了现有技术的范围（径向数据除外）。其次，主要研究者希望测试迄今为止开发的工具箱针对新困难的稳健性，例如关键正则性不对应于（强制）守恒量的问题或对称性破缺的问题。 该项目的最后一部分涉及低正则空间中初始数据（修改后的）Korteweg-de Vries 方程的全局适定性问题。由于完全可积技术，这个问题在周期性情况下比在非周期性情况下更容易理解。 首席研究员建议从纯偏微分方程的角度重新审视卡佩勒和托帕洛夫的这些新进展，希望找到一种合适的规范，能够以低正则性处理非周期性情况。本项目中研究的方程有着丰富的历史。它们受到了数学家和物理学家的研究，因为它们捕捉了某些物理行为的重要方面，同时保持了有吸引力的简单性。 因此，它们成为研究偏微分方程的新分析技术的滋生地。尽管要研究的方程相对于科学或工业的需求来说过于简单，但主要研究者相信，对这些方程的研究将促进具有更广泛适用性的工具的开发，而即使是最微小的加速进入可以处理诸如著名的纳维-斯托克斯方程等超临界方程的时代，也确实是非常有益的。 与工具箱的开发并行的是它的传播。 首席研究员将继续朝这个方向开展活动，包括维护一套关于该材料的讲义。