Prandtl方程整体解的存在惟一性

Abstract

考虑非定常的Prandtl方程U(t,x)=xmU1(t,x),且m≥1,0≤x<L的特殊情况,在本文的条件下,所研究的方程具有奇性.首先利用Crocco变换把Prandtl方程变换成一个关于w的方程,然后将其正则化,借助于正则化以后的方程得到wε(正则化后方程的解)及其各种一阶导数的估计.利用得到的各种估计通过取极限得到了Crocco变换后方程解的存在惟一性.最后返回边界层,得到Prandtl方程全局解的存在惟一性.

The Prandtl system for a non-stationary Prandtl equation is considered in this paper.It is assumed that U(t,x,y)=x~mU_1(t,x) in the nonstationary case,where m≥1 and 0≤x≤L.The aim of this article is to prove the existence and uniquence of the globle solution to this equation.But it is a singular equation in the condition of this article.Firstly the Prandtl equation is changed to another equation of w by Crocco transformation.Then the equation is regularized and some estimates of w_ε(the solution of the regularlized equation) and the partial of w_ε are gotten.The limit of w_ε can be proved to be the solution of the equation of w.Lastly the existence and uniquence of the globle solution to Prandtl equation is proved.