On well-posedness and regularity properties for fluid equations and nonlinear dispersive equations

流体方程和非线性色散方程的适定性和正则性

• 批准号: 0758247
• 负责人: Natasa Pavlovic
• 金额: $ 12.68万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0758247&HistoricalAwards=false
• 关键词: well; posedness; regularity; properties; fluid

Pavlovi'c plans to gain better understanding of fluid and nonlinear wave and dispersive equationsvia analyzing dispersive equations using harmonic analysis techniques that turned out to be usefulin the context of fluid equations and vice verse. The first group of proposed problems focuses onthe Navier-Stokes equations that describe fundamental properties of viscous fluids. One approachin studying existence of solutions to the Navier-Stokes is to obtain solutions to the correspondingintegral equation. Existence of these solutions in 3D has been proved only locally in time andglobally for small initial data. Hence it is important to understand behavior of these solutions in"critical spaces" that preserve scaling invariance. With her collaborators the PI will investigate thebehavior of solutions to the Navier-stokes equations in critical spaces. Questions of interest includestability of self-similar solutions in critical spaces (which is motivated by the analogy with solitonsin the context of dispersive equations) and a long standing open problem related to well-posednessof the Navier-Stokes equations in the largest critical space. The second group of proposed problemsconcentrates on nonlinear dispersive equations. Many of the important structural properties(e.g. conserved or monotone quantities) of the nonlinear Schrodinger equations (NLS) are at lowregularities, and to exploit these features one needs to establish existence theory at low regularities.Pavlovi'c proposes to continue her work on establishing global well-posedness for certain class ofNLS equations corresponding to low regularity data. The PI will employ and further investigatetools that were useful in recent advances in the field, such as interaction Morawetz estimates.The third group of problems is related to super-critical nonlinear wave and NLS equations. Heresuper-critical refers to equations with conserved quantities at lower regularities than the scalinginvariant norm (the 3D Navier-Stokes is an example). Motivated by her earlier work with Katzon partial regularity of the Navier-Stokes equations, Pavlovi'c proposes to use microlocalizationtechniques in order to obtain a partial regularity result for super-critical NLS and wave equations.Suggested problems involve important mathematical questions such as existence and regularityof solutions to nonlinear PDEs that describe motion of fluid or various wave phenomena. Forinstance, the theory of the Navier-Stokes equations in three dimensions is far from being complete.The outstanding open problems, whose better understanding would have impact in the fields fromoceanography to cosmology, are global existence, uniqueness and regularity of smooth solutions tothe Navier-Stokes in 3D. On the other hand, the NLS and their combinations with the Kortewegde-Vries and wave equations have been proposed as models for many basic wave phenomena.Such a physical relevance of the equations motivates mathematical explorations. The proposedactivity seeks to find an interdisciplinary approach to questions arising from fluid and dispersivePDEs. In particular, the PI plans to analyze dispersive equations using sophisticated techniquesof harmonic analysis that turned out to be useful in the context of fluid equations and vice verse.

巴甫洛维奇计划通过使用调和分析技术分析色散方程来更好地理解流体和非线性波以及色散方程，这些技术在流体方程中非常有用，反之亦然。第一组提出的问题集中于描述粘性流体基本性质的纳维-斯托克斯方程。研究纳维-斯托克斯解的存在性的一种方法是获得相应积分方程的解。这些解在 3D 中的存在仅在局部时间上得到证明，而对于较小的初始数据则在全局上得到证明。因此，了解这些解决方案在保持缩放不变性的“关键空间”中的行为非常重要。 PI 将与她的合作者一起研究临界空间中纳维-斯托克斯方程解的行为。感兴趣的问题包括临界空间中自相似解的稳定性（这是由色散方程背景下与孤子的类比引起的）以及与最大临界空间中纳维-斯托克斯方程的适定性相关的长期悬而未决的问题。第二组提出的问题集中在非线性色散方程上。非线性薛定谔方程（NLS）的许多重要结构性质（例如守恒量或单调量）都具有低规律性，要利用这些特征，需要建立低规律性的存在理论。Pavlovi'c 建​​议继续为对应于低规律性数据的某类 NLS 方程建立全局适定性的工作。 PI 将采用并进一步研究该领域最新进展中有用的工具，例如相互作用 Morawetz 估计。第三组问题与超临界非线性波和 NLS 方程有关。这里，超临界是指具有比标度不变范数更低的规律性的守恒量的方程（3D 纳维-斯托克斯就是一个例子）。受她早期对纳维-斯托克斯方程 Katzon 部分正则性研究的启发，Pavlovi'c 提出使用微定位技术来获得超临界 NLS 和波动方程的部分正则性结果。提出的问题涉及重要的数学问题，例如描述流体运动或各种波动现象的非线性偏微分方程解的存在性和正则性。例如，三维纳维-斯托克斯方程的理论还远未完善。 3D 纳维-斯托克斯光滑解的全局存在性、唯一性和规律性是突出的开放问题，其更好的理解将对从海洋学到宇宙学的领域产生影响。另一方面，NLS 及其与 Kortewegde-Vries 和波动方程的组合已被提出作为许多基本波动现象的模型。方程的这种物理相关性激发了数学探索。拟议的活动旨在寻找一种跨学科的方法来解决由流体和色散偏微分方程引起的问题。特别是，PI 计划使用复杂的调和分析技术来分析色散方程，这些技术在流体方程的背景下非常有用，反之亦然。