轴对称Navier-Stokes方程组的整体适定性研究

The three-dimensional Navier-Stokes system is important in partial differential equations and fluid dynamics. The regularity of its Leray-Hopf solution is one of the Millionaire prize problems. This project will consider the global regularity problem when the initial data are axisymmetric and with swirl. We shall consider the following five aspects: first, find more precise a priori estimate; second, research on the weighted regularity criteria involving only one component of the velocity or the vorticity, in the framework of Lebesgue or Besov spaces, so as to find more special global well-posedness results; third, view the system as a two-dimensional one, study the controllability of the convective terms; fourth, consider the elliptic cylindrical transformation, find how eccentricity impacts on the global regularity; fifth, extend the results of homogeneous case to the inhomogeneous one, and see how the inhomogeneity impacts on the global regularity problem. All in all, we study the project in five dimensions: a priori estimates, regularity criteria/small-data global existence, two-dimensional view point, balance between two and three dimensions and possible extension to inhomogeneous case.

三维不可压Navier-Stokes方程组是偏微分方程和流体动力学的重要方程组,其Leray-Hopf弱解的正则性是千禧年大奖难题之一.本项目拟就初值是轴对称且角向速度不为零时讨论其整体适定性.本项目拟从以下五个方面来研究:第一,进一步探讨更多/更精确的先验估计;第二,在Lebesgue或Besov空间的框架下深入研究依赖于速度或旋度的一个分量的带权型正则性准则,从而获取更多的特殊情形下的整体适定性结果;第三,把方程看成二维非线性偏微分方程,探讨对流项的可控性;第四,考虑椭圆柱坐标变换,看离心率满足什么条件时能获取整体适定性,探讨二维与三维轴对称之间的一个平衡;第五,将齐性情形推广到非齐性情况,看非均匀指标满足什么条件时亦能有整体适定性.综上,我们从先验估计,正则性准则/小性整体解,二维方程观点,二维三维的平衡,推广到非齐性情形五个方面进行多维度深入研究.

本项目主要研究轴对称 Navier-Stokes 方程组, 探讨解的先验估计, 解的正则性准则与解的整体适定性. 发表相关 SCI 一作论文 22 篇. 得到了轴对称 Navier-Stokes 方程组关于旋度角向分量的带权型先验估计; 基于旋度角向分量的带权型正则性准则; 速度或者旋度部分分量的带权型最优准则; 轴对称磁流体方程组的对流项的优良性质及正则性准则; 带 Bernard 效应的磁流体方程的整体适定性. 给出了轴对称流体运动性态中我们有的 (先验估计), 及我们要的 (正则性准则). 此外, 轴对称流体的整体适定性也在某些特殊情形下得到了.