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.

巴甫洛夫计划通过使用谐波分析技术来分析色散方程，从而更好地理解流体和非线性波和色散方程，这种技术在流体方程的背景下被证明是有用的，反之亦然。提出的第一组问题集中在描述粘性流体基本性质的Navier-Stokes方程上。研究Navier-Stokes方程解的存在性的一种方法是求得相应的积分方程的解。对于小的初始数据，这些解的存在性仅在局部时间和全局范围内得到了证明。因此，理解这些解决方案在保持尺度不变性的“关键空间”中的行为是很重要的。PI将与她的合作者一起研究临界空间中Navier-stokes方程解的行为。感兴趣的问题包括临界空间中自相似解的可包容性（这是由色散方程中孤子的类比所激发的）以及与最大临界空间中Navier-Stokes方程的适定性相关的长期开放问题。第二组提出的问题集中于非线性色散方程。许多重要的结构特性(例如：非线性薛定谔方程（NLS）的守恒量（单调量）是低规律的，为了利用这些特征，需要建立低规律的存在性理论。Pavlovi'c建议继续她的工作，为一类对应于低正则性数据的nls方程建立全局适定性。PI将采用并进一步研究在该领域最新进展中有用的工具，例如Morawetz估计的交互作用。第三组问题与超临界非线性波动方程和NLS方程有关。这里的“超临界”指的是守恒量的规律性低于标度不变范数的方程（3D Navier-Stokes就是一个例子）。受她早期研究Navier-Stokes方程的Katzon部分正则性的启发，Pavlovi'c建议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。建议的问题涉及重要的数学问题，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规律性。例如，三维的Navier-Stokes方程的理论还远远不够完善。其中最突出的开放性问题是三维Navier-Stokes光滑解的全局存在性、唯一性和规律性，对这些问题的更好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已被提出作为许多基本波动现象的模型。这种方程的物理相关性激发了数学探索。拟议的活动旨在找到一种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计划使用复杂的谐波分析技术来分析色散方程，这种技术在流体方程的背景下很有用，反之亦然。