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

巴甫洛维奇计划通过使用调和分析技术分析色散方程，从而更好地了解流体、非线性波和色散方程，事实证明，这种技术在流体方程中很有用，反之亦然。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N-S方程解的存在性的一种方法是求解相应的积分方程解。对于较小的初始数据，这些解的存在仅在时间上是局部的，在全局上是被证明的。因此，了解这些解在保持比例不变性的“临界空间”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(这是由色散方程中的孤子的类比而引起的)和一个与最大临界空间中的Navier-Stokes方程的适定性有关的长期未解决的问题。第二组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(如守恒量或单调量)都是低正则的，要利用这些性质需要建立低正则下的存在性理论。Pavlovi‘c建议继续她的工作，建立与低正则数据相对应的某类NLS方程的全局适定性。PI将使用和进一步研究在该领域的最新进展中有用的工具，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守恒量的方程(3D Navier-Stokes就是一个例子)。受早期研究Navier-Stokes方程的Katzon部分正则性的启发，Pavlovi‘c建议使用微区化技术来获得超临界NLS和波动方程的部分正则性结果。所建议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Navier-Stokes方程的理论还很不完善，对三维Navier-Stokes方程光滑解的整体存在性、唯一性和正则性的理解将对从地形学到宇宙学的各个领域产生重大影响。另一方面，NLS及其与Kortewegde-Vries和波动方程的组合被提出作为许多基本波动现象的模型。这些方程的物理相关性激发了数学探索。拟议的活动试图找到一种跨学科的方法来解决流动和分散的PDE所产生的问题。特别是，PI计划使用复杂的调和分析技术来分析色散方程，事实证明，这种技术在流体方程中很有用，反之亦然。

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.