一类线性方程组的特征边界层分析

Analysis of Characteristic Boundary Layers for a Class ofLinear Equations

本文主要研究的是一维线性粘性抛物方程与无粘双曲方程之间的解的渐近极限。我们假定相应的无粘方程的边界是特征的,去研究粘性解与无粘解之间的关系。我们用渐近展开的方法讨论不同区域内粘性方程的近似解,并利用加权能量估计的方法讨论Prandtl 型的边界层方程解的存在性, 以证明边界层的稳定性。通过对近似解与粘性问题真实解之间的误差进行估计,我们最终得到粘性方程的解与无粘解的渐近等价关系。

In this paper, we mainly study the asymptotic limit of the solution of the initial boundary value problem for one-dimensional linear equations. We assume that the boundary of the corresponding inviscid equation is characteristic, and study the re- lationship between the viscous solution and the inviscid one. The boundary layer is characteristic. We use the method of matched asymptotic expansions to discuss the approximate solution of viscous equation in different domains. By using the method of weighted energy estimates, we obtain the existence of solutions for Prandtl type boundary layer equations. In order to prove the stability of the boundary layer, the error between the approximate solution and the real solution of the viscous problem is estimated. Finally, we obtain the asymptotic equivalence between the solutions of the viscous equation and the inviscid one.