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谐波分析技术分析色散方程来更好地理解流体和非线性波和色散方程，谐波分析技术在流体方程中是有用的，反之亦然。第一组提出的问题集中在描述粘性流体基本性质的Navier-Stokes方程。研究Navier-Stokes方程解的存在性的一种方法是求出相应积分方程的解。在3D中，这些解的存在性已被证明仅在局部时间和全球小的初始数据。因此，重要的是要了解这些解决方案在“临界空间”，保持标度不变性的行为。PI将与她的合作者一起研究临界空间中Navier-Stokes方程的解的行为。感兴趣的问题包括临界空间中自相似解的稳定性（这是由色散方程中孤立子的类比所激发的）和一个长期存在的与最大临界空间中Navier-Stokes方程的适定性有关的开放问题。第二组提出的问题集中在非线性色散方程。非线性薛定谔方程（NLS）的许多重要结构性质（如守恒量或单调量）都是在低正则性条件下的，为了利用这些性质，人们需要建立低正则性条件下的存在性理论. Pavlovic提出继续建立某些类型的NLS方程的整体适定性. PI将采用并进一步研究在该领域的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。这里的超临界是指守恒量低于标度不变范数的方程（3D Navier-Stokes方程就是一个例子）。Pavlovi 'c受她早期研究Navier-Stokes方程Katzon部分正则性的启发，提出使用微局部化技术来获得超临界NLS和波动方程的部分正则性结果，所提出的问题涉及到一些重要的数学问题，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Navier-Stokes方程的理论还远未完善，三维Navier-Stokes方程的整体解的存在性、唯一性和正则性是一个悬而未决的问题，对这些问题的深入理解将对从海洋学到宇宙学的各个领域产生影响。另一方面，NLS及其与Kortewegde-Vries方程和波动方程的组合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。拟议的活动旨在找到一个跨学科的方法来解决流体和分散的偏微分方程所产生的问题。特别是，PI计划使用谐波分析的复杂技术来分析色散方程，谐波分析在流体方程中非常有用，反之亦然。