三维非匀质Navier-Stokes方程的部分正则性

Partial regularity of the 3-D inhomogeneous Navier-Stokes equations

本文首先引进非匀质Navier-Stokes方程恰当弱解的概念.当初始密度接近正常数的情形时,通过结合局部能量不等式、Sobolev嵌入、压力估计和blow up分析技术,本文建立了恰当弱解的内部正则性准则.最后利用内部正则性准则证明了恰当弱解可能奇异点集的一维Hausdorff测度为零.

In this paper, we first introduce the concept of suitable weak solutions for the inhomogeneous Navier- Stokes equations. In the case when the initial density is close to a positive constant, by combining local energy inequality, Sobolev embedding, pressure estimate and blow-up analysis, we establish interior regularity criterion of suitable weak solutions. Finally, we show that the 1-D Hausdorff measure of the set of possible singular points of suitable weak solutions vanishes by applying the interior regularity criterion.