轴对称Navier-Stokes方程解的性态研究

Study on properties of solutions of the Navier-Stokes(NS) equations has always been a fairly important topic in partial differential equations. Under the background that the regularity of weak solutions of the unsteady NS equations and the vanishing of D-solutions of the steady NS equations haven't been solved completely, our project raises the following topic: Investigating the regularity and vanishing of solutions of a special type of(axially symmetric) NS equations by studying the upper-bound estimate of solutions of the unsteady NS equations and the decay rate of D-solutions of the steady NS equations. Problems of lack of the accuracy of the upper-bound estimate of solutions of the unsteady NS equations and the decay rate of D-solutions of the steady NS equations will be overcome by using techniques of the localized Biot-Savart law formula, scaling invariance, iterations and odd-even extensions etc. At last, we aim at proving the regularity of weak solutions of the unsteady axially symmetric NS equations and the vanishing of D-solutions of the steady axially symmetric NS equations under suitable and reasonable conditions. The innovation of our project are:1,Study the regularity and decay rate of solutions by using the scaling invariance of NS equations;2.Introduction of Green functions on a special domain with suitable constraints;3.Application of dimensionality reduction.

对Navier-Stokes(NS)方程解的性态的研究一直都是偏微分方程中比较重要的课题。在目前非定常NS方程弱解正则性和定常NS方程D解的消失性问题仍未得到完全解决的背景下，本项目提出通过研究非定常NS方程解的上界估计和定常NS方程D解的衰减性估计的方法来研究一类特殊的(轴对称)NS方程解的正则性和消失性的课题；拟采用局部化的Biot-Savart law形式，NS方程解伸缩的尺度不变性，重复迭代，奇偶延拓等技术手段来克服非定常NS方程中解上界估计不够精确和定常NS方程D解的衰减率不够的问题；最终达到在适当的合理的限制条件下，完成非定常轴对称NS方程弱解正则性和定常轴对称NS方程D解的消失性的理论证明的目标。本项目的创新之处在于:1.采用NS方程解伸缩不变性的特性来研究解正则性和衰减性；2.引入了在一类特殊区域上满足一定条件的Green函数;3.降维思想的运用。

本项目研究主要内容,重要结果及其科学意义如下：. (1). 研究了轴对称Navier-Stokes方程D解速度以及涡度在空间无穷远处的衰减率，优化了之前结果。. (2). Navier-Stokes方程在平行板之间以及无限长管道之间D解的刘维尔定理。在平行两板之间，在非滑移的物理边界条件下，我们证明了D解的消失性，此结果较之前文献结果主要在于我们对D解不再做其他先验假设。在两个周期变量的无限长管道中，我们可以证明在部分速度分量Dirichlet积分有限，其他分量甚至可以增长的条件下解的刘维尔定理。 . (3). 证明了磁流体方程组中，两板之间各种边界条件下D解的刘维尔定理。. (4). Navier-Stokes方程古代解的平凡性证明。我们在最优增长条件下证明了无旋古代解的刘维尔定理，此结果改进了之前关于解的有界性假设。