Prandtl方程整体解的存在惟一性

The Existence and Uniquence of the Globle Solution to Prandtl Equation

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^mU1 （t,x） in the nonstationary case,where m≥1 and ≤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 ω4 （the solution of the regularlized equation） and the partial of w, are gotten. The limit of w, 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.