稳态不可压Navier-Stokes方程解的存在性及衰减性研究

Starting from the pioneering works by Leray in 1933, the study on the existence and decay properties of weak solutions to the flux problem for the steady incompressible Navier-Stokes equations on some multiple connected domains is always a very active research area in fluid dynamics. This model was used to describe some important physical phenomena likes the flow passing through some obstacles (like bridge pier), the wake flow appeared after a boat moving in the lake. The applicant and his collaborator have proved some Liouville type theorems for steady Navier-Stokes equations and also obtained some interesting decay rates estimates in the axisymmetric case. Based on these works, this program will concentrate on the following three closely related problems: 1) the existence of weak solutions to the steady Navier-Stokes equations in some multiple connected domains which are helical invariant or symmetric with respect to a plane or two perpendicular planes; 2) for the axisymmetric Navier-Stokes equations, we will investigate the possible decay rates at infinity especially in the direction along the axis and also the asymptotic expansion; 3)based on the study of the axisymmetric case, we are interested on the possible self-improvement decay mechanism that maybe embedded in the steady Navier-Stokes equations.

关于稳态不可压Navier-Stokes方程在多连通区域上流量问题弱解的存在性及衰减性的研究自1933年Leray的开创性工作以来一直是流体力学数学研究中十分活跃的一个分支。该模型一般用来描述水流经某障碍物（如桥墩）形成的绕流、船在湖泊中航行时船体后方形成的尾流（wake）等现象。申请人和合作者已经证明了稳态Navier-Stokes方程的一些消失性准则及轴对称情形解在无穷远处的衰减速率估计。在此基础上，本项目拟研究以下三个紧密相关的问题：1）稳态Navier-Stokes方程螺旋不变或关于某一平面或某两个相互垂直的平面对称的多连通区域上流量问题弱解的存在性；2）研究三维轴对称稳态Navier-Stokes方程Leray解在无穷远处特别是关于对称轴方向的衰减速率估计，以及解在无穷远处的渐近展开；3）考察三维稳态Navier-Stokes方程Leray解在衰减性方面可能存在的自我改进机制。

本项目主要研究稳态不可压Navier-Stokes方程在多连通区域上非齐次边值问题弱解的存在性及衰减性。在项目资助期间，项目主持人与合作者改进了三维稳态轴对称Navier-Stokes方程有限Dirichlet能量解的二阶导数的衰减速率；同时证明螺旋不变的多连通区域上非齐次边值问题有限Dirichlet能量弱解的存在性；对于轴对称不可压磁流体方程组，利用方程特有的结构及Brezis-Gallout不等式、尺度变换和加权能量估计等技巧研究了有限Dirichlet能量强解在无穷远处尽可能优的衰减速率。项目主持人与合作者证明了无穷长轴对称管道中轴对称带非零旋涡的光滑亚音速流的整体适定性；项目主持人与合作者发现了三维欧拉方程组的双曲椭圆耦合结构关于形变张量与旋度的分解，解决了速度场正则性较压强缺失一阶导数的问题，并将这一思想推广到三维定常Euler-Poisson方程组，证明了三维长方体管道内满足适当物理边界条件的光滑亚音速流的存在唯一性。项目主持人与合作者还证明三维张口管道中跨音速激波关于管道壁轴对称扰动和非零旋涡扰动、超音速来流及出口压力的轴对称扰动的结构稳定性，为进一步研究一般三维管道中跨音速激波的结构稳定性奠定基础。项目执行期间正式发表SCI文章4篇，有1篇接受发表，另有两篇文章在投稿中。在项目资助期间先后招收培养硕士研究生2人，博士研究生2人。