Dispersive PDE at critical regularity

临界正则性的色散偏微分方程

• 批准号: 0965029
• 负责人: Monica Visan
• 金额: $ 15.4万
• 依托单位: University of California-Los Angeles
• 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: 2009-07-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0965029&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.

该奖项是根据2009年《美国回收与再投资法》（公法111-5）资助的。该项目的主要目的是进一步了解解决方案对某些分散方程的全球/大数据行为的理解。更确切地说，主要研究者认为非线性Schrodinger，Klein-Gordon和（修改的）Korteweg-de Vries方程的全球适应性和散射问题，用于属于关键/低规范性Sobolev空间的初始数据。在过去的几年中，非线性波（NLW）和Schrodinger（NLS）方程的临界定型问题引起了极大的关注。这些作品开发了一套强大的工具和技术，旨在以保守的关键规律性解决NLW和NLS。该项目的主要目的是加强和扩展该工具箱。直接的目标包括处理焦点（低维）临界NLS和散落/聚焦的质量关键NLS，这些问题可能超出了现有技术的影响范围（在径向数据的情况下除外）。其次，主要研究人员希望测试迄今为止在新困难方面开发的工具箱的鲁棒性，例如，关键规律性与（强制性）保守数量或损坏的对称性问题相对应的问题。 该项目的最后一部分与（修改的）Korteweg-de Vries方程相关的全球适应性问题，用于低规律性空间中的初始数据。得益于完整的集成性技术，在周期性的情况下，该问题比在非周期性的情况下更好地理解了。 首席研究人员建议从纯粹的部分微分方程的角度来重新审视这些新的进步，以期发现一个适当的规格，以允许在低规律性的情况下对非周期案例进行处理。数学家和物理学家都对它们进行了研究，因为它们捕获了某些身体行为的重要方面，同时保持了吸引人的简单性。 因此，它们是用于研究部分微分方程的新分析技术的繁殖场。尽管相对于科学或工业的需求，要进行的研究方程式大大简化了，但主要研究人员认为，对这些方程式的研究将促进具有更广泛适用性的工具的开发，而当诸如著名的Navier-Stokes方程之类的超级临界方程式也可以很好地处理时，即使是最小的时代也可以非常有益。 与工具箱的开发平行是其传播。 首席调查员将继续朝这个方向进行活动，包括维护有关此材料的一系列讲座。