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.

Pavlovi'C计划通过使用谐波分析技术来更好地了解流体和非线性波和分散方程，从而分析分散方程，这些技术证明在流体方程和副诗句的上下文中很有用。第一组提出的问题集中于描述粘性流体基本特性的Navier-Stokes方程。在研究Navier-Stokes的解决方案的一种方法中，是获得相应综合方程的解决方案。对于小初始数据而言，仅在及时和全球方面仅在本地证明了这些解决方案的存在。因此，重要的是要在保留缩放不变性的“关键空间”中了解这些解决方案的行为。 PI与她的合作者一起研究了关键空间中Navier-Stokes方程的解决方案的行为。感兴趣的问题包括在关键空间中自相似解决方案的性能（这是由与孤子的类似环境类比的相比，以及与最大关键空间中Navier-Stokes方程相关的长期存在的开放问题。第二组提出的问题对非线性分散方程式进行了浓缩。非线性Schrodinger方程（NLS）的许多重要结构特性（例如，保守或单调量）处于较低的状态，为了利用这些特征，人们需要在低规律性下建立存在理论。Pavlovi'c建议继续她的工作来建立她的工作，以建立她的工作与低规律性数据相对应的某些类别的NLS方程的全球适应性。 PI将采用和进一步的研究中的研究，这些研究在该领域的最新进展中有用，例如Morawetz估计值。第三组问题与超临界非线性波和NLS方程有关。 Heresuper-关键是指在规律较低的情况下保守数量的方程式（3D Navier-Stokes就是一个例子）。 Pavlovi'C提出了以前与Katzon部分规律性的局部规律性的促进的，Pavlovi'c建议使用微钙化技术，以获得超临界NLS和波浪方程的部分规律性。针对非线性PDE的定期溶液，描述了流体或各种波浪现象的运动。 forinstance，在三个维度上的Navier-Stokes方程理论远非完整。杰出的开放问题（其更好的理解都会影响到范围摄影对宇宙学的领域，是全球的存在，平滑解决方案的独特性和规律性。 3d中的stokes。另一方面，已经提出了NLS及其与Kortewegde-Vries和Wave方程的组合作为许多基本波浪现象的模型。方程的物理相关性激发了数学探索。提议的活动试图找到一种跨学科的方法来解决由流体和分散pdes引起的问题。特别是，PI计划使用谐波分析的复杂技术分析分散方程，这些技术在流体方程式和恶习诗句的背景下很有用。