三维Navier-Stokes方程组解的正则性相关问题的一些探索  

Some problems on the regularity of solutions for the 3D Navier-Stokes equations

作　　者：方道元[1] 张挺[1] 

机构地区：[1]浙江大学数学科学学院,杭州310027

出　　处：《中国科学：数学》2018年第1期45-70,共26页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11671353);浙江省自然科学基金(批准号:LR17A010001)资助项目

摘　　要：本文回顾了近年来作者团队对三维不可压缩Navier-Stokes方程组Cauchy问题所作的一些探索.众所周知,三维不可压缩Navier-Stokes系统存在整体Leray-Hopf弱解.当弱解满足Prodi-Serrin条件时,解是正则的.本文在解正则性条件的判别方面取得了一些新结果.特别对于轴对称系统,当旋转速度为零时,系统的整体适定性结论是众所周知的.本文在研究中发现了一个新的守恒量,进而得到了旋转速度非零时其轴对称解正则性条件的一些新进展,还得到了一个系统只要求初始旋转速度小的整体适定性结果,进一步还将结果推广到变密度的系统.最后,考虑了一类超耗散广义Navier-Stokes系统的整体适定性,其中水平黏性项具有更高阶导数D_h^(2α),α≥4/3.In this paper, we review some recent work on the Cauchy problem of the three-dimensional incompressible Navier-Stokes equations in recent years. It is well known that the three-dimensional incompressible Navier-Stokes system has the global Leray-Hopf weak solutions. When the weak solution satisfies the Prodi- Serrin condition, the solution is regular. We obtain some new results in the regularity conditions. Especially, for axisymmetric system, when the rotation speed is zero, it is well known that the system is globally well-posed. We find a new conserved quantity, and obtain some new advances in the regularity conditions of axisymmetric solutions with nonzero swirl, also get the global well-posedness results with the small initial rotating speed. Furthermore, we also get the similar result for the inhomogeneous system. In the end, we consider the global well-posedness of a class of generalized Navier-Stokes systems with the higher order horizontal viscosity Dh^2α,α≥4/3.

关 键 词：不可压缩Navier-Stokes方程组 正则性条件 整体适定性 

分 类 号：O175[理学—数学]