稳态可压Euler方程组的形变张量与旋度分解  

A deformation-curl decomposition for three dimensional steady Euler equations

作　　者：翁上昆 辛周平[2] Shangkun Weng;Zhouping Xin

机构地区：[1]武汉大学数学与统计学院武汉大学数学协同创新中心,武汉430072 [2]香港中文大学数学科学研究所

出　　处：《中国科学：数学》2019年第2期307-320,共14页Scientia Sinica：Mathematica

基　　金：国家自然科学青年基金(批准号:11701431);中央高校基本科研经费(批准号:201-413000047);中组部青年千人项目(批准号:212100004);香港研究局(批准号:CUHK 4048/13P和CUHK-14305315);国家自然科学基金委员会和香港研究局联合资助基金(批准号:N-CUHK443-14)资助项目

摘　　要：本文提出了稳态Euler方程组关于形变张量与旋度的一个分解,这个分解的关键想法在于如下简单的观察:注意到利用Bernoulli定律,可以将密度改写为Bernoulli函数、熵和速度这三者的代数表示式,从而将质量守恒方程改写成某一对称矩阵与形变张量的矩阵Frobenius内积形式.进一步利用动量守恒方程组,我们发现旋度可以由一个输运方程以及两个关于Bernoulli函数和熵的代数方程表示出来.最后利用形变张量方程与旋度方程求解速度场和密度.这个分解的好处在于,我们最后构造的解的速度场与压力、Bernoulli函数以及熵有相同的正则性.基于这个分解,本文证明了有限长方体管道中满足适当物理边界条件的亚音速解的存在唯一性.In this paper, we provide a deformation-curl decomposition of three dimensional steady Euler equation. The key issue in this new decomposition is based on a simple observation that the density can be represented as a function of the Bernoulli’s function, the entropy and the speed, and hence the density equation can be rewritten as a Frobenius inner product of a symmetric matrix with the deformation matrix: A(u) : D(u) = 0, where D(u) =(1/2(?_juk + ?_ku_j))_j^3,k=1is the deformation matrix. Furthermore, employing the momentum equations,we find the vorticity can be resolved by a transport equation with another two algebraic equations involving the Bernoulli’s function and entropy. We also obtain the velocity by solving the deformation and curl system.The most important advantage of this decomposition is the velocity, the Bernoulli’s function and entropy we obtained share the same regularity as the pressure or density. Based on this decomposition, we prove the existence and uniqueness of subsonic flows in a finitely long rectangular nozzle by prescribing suitable physical boundary conditions.

关 键 词：稳态Euler方程组 双曲椭圆耦合 形变张量 旋度 相容性条件 

分 类 号：O175[理学—数学]