纳维-斯托克斯方程解正则性的局部标准

IF 2.4 2区 数学 Q1 MATHEMATICS
Congming Li , Chenkai Liu , Ran Zhuo
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引用次数: 0

摘要

Ladyzhenskaya-Prodi-Serrin型Ls,r准则是不可压缩纳维-斯托克斯方程组解的正则性准则,是研究克莱数学研究所提出的不可压缩N-S方程组千年难题的基础。这种全局 Ls,r 准则通常很大,因此很难控制。用某种局部规范代替全局 Ls,r 规范是很有意思的。在这篇文章中,我们引入了局部 Ls,r 空间,并为方程解的正则性建立了一些局部标准。事实上,我们得到了方程解的一些先验估计值,这些估计值只取决于某些局部 Ls,r 型规范。对于合理的初始值来说,这些局部规范很小,而对于全局正则解来说,这些局部规范仍然很小。因此,推导出局部 Ls,r 型规范的微小性甚至有界性,是肯定地回答千年问题的必要条件和充分条件。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A localized criterion for the regularity of solutions to Navier-Stokes equations

The Ladyzhenskaya-Prodi-Serrin type Ls,r criteria for the regularity of solutions to the incompressible Navier-Stokes equations are fundamental in the study of the millennium problem posted by the Clay Mathematical Institute about the incompressible N-S equations. This global Ls,r norm is usually large and hence hard to control. Replacing the global Ls,r norm with some kind of local norm is interesting. In this article, we introduce a local Ls,r space and establish some localized criteria for the regularity of solutions to the equations. In fact, we obtain some a priori estimates of solutions to the equations depend only on some local Ls,r type norms. These local norms, are small for reasonable initial value and shall remain to be small for global regular solutions. Thus, deriving the smallness or even the boundedness of the local Ls,r type norms is necessary and sufficient to affirmatively answer the millennium problem.

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来源期刊
CiteScore
4.40
自引率
8.30%
发文量
543
审稿时长
9 months
期刊介绍: The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. Research Areas Include: • Mathematical control theory • Ordinary differential equations • Partial differential equations • Stochastic differential equations • Topological dynamics • Related topics
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