分数次不可压缩Navier-Stokes方程的正则性理论

This project will be mainly focused on regularity theory of the fractional incompressible Navier-Stokes equations. With the new methods developed in our previous work, we will study weak solutions to the fractional incompressible Navier-Stokes equations. Firstly we will study partial regularity of suitable weak solutions to the high-order fractional case (1<s<5/4)，and give estimates on Hausdorff dimension of singular set of suitable weak solutions. Our goal is to extend partial regularity result given by Caffarelli-Kohn-Nirenberg of the classical 3-dimentional incompressible Navier-Stokes equations. Secondly, the general weak solutions will be also considered. We hope to give some results similar to the Serrin-type regularity criterion in classical cases and prove that when any weak solution satisfies some integrability condition, it would be smooth. Finally,we will consider the regularity of weak solutions to the equations with critical exponents (non-steady case s=4/3 and steady case s=1/2 ). Since the method used in our previous work does not work for this cases, we plan to develop a new estimate and technique to get partial regularity of suitable weak solutions.

本课题将重点讨论分数次Navier-Stokes方程的正则性理论。本项目拟基于申请人先前工作的新方法, 研究此类方程弱解的正则性。首先，对高阶分数次情 形(1 < s < 5/4 )，我们研究适当弱解的部分正则性, 并对适当弱解奇异集的Hausdorff维数进行估计。我们的目标是证明奇异集的(5-4s)-维Hausdorff测度为零，从而推广Caffarelli-Kohn-Nirenberg 关于经典3维不可压缩Navier-Stokes方程部分正则性结果。其次，我们还将研究方程一般弱解的正则性。我们希望给出类似经典情形Serrin-型正则性准则的结果，证明当弱解满足一定的可积条件时，它就是光滑的。最后，我们会探讨临界指标情形(非静态: s=3/4及静态: s=1/2)的方程适当弱解的正则性。由于先前的方法并不适用，我们计划发展一套新的估计和方法来得到适当弱解的部分正则性。

本项目主要研究分数次Navier-Stokes方程弱解的正则性理论, 这是当今非线性偏微分方程领域的核心问题，在流体力学与随机分析等领域有着重要的应用。在我们的工作中，我们得到了静态情形方程弱解的部分正则性结果。具体说来，我们建立了适当弱解的正则性准则，并对适当弱解奇异集的Hausdorff维数得到了精确地估计。此外，我们还研究了一般化的Monge-Ampère方程，给出了弱解的比较原理。