一类具有多个激波层结构粘性守恒律方程的粘性消失极限问题

Vanishing Viscosity Limit for a Class of Viscous Conservation Laws with Multiple Shock Layers

本文研究的是带有两个无相互作用的激波层的一维拟线性粘性方程的柯西问题解的渐近极限。目的是理解无相互作用的粘性激波层的进化与构造以及外部无粘双曲流之间的相互作用,井证明粘性解在远离激波层区域中一致收敛于分片光滑的无粘解,这是基于匹配渐近分析法和能量估计法。文章先利用匹配渐近展开的方法构造粘性方程的近似解,再利用能量估计的方法估计近似解与粘性方程真实解之间的误差,得到误差的H1估计,并用Sobolev嵌入得到L∞估计,从而证明两类方程的渐近等价性。

In this paper, we study the asymptotic limit of the solution of the Cauchy problem for one-dimensional quasilinear viscous equation with two non-interacting shock layers. The aim is to understand the evolution and construction of the non-interaction viscous shock layer and the interaction between the external inviscid hyperbolic flow, and to prove that the viscous solution converges uniformly to the piecewise smooth inviscid solution in the region far from the shock layer. This is based on the method of matched asymptotic expansions and energy estimates. Firstly, the approximate solu- tion of the viscous equation is constructed by using the method of matched asymptotic expansions, and then the error between the approximate solution and the real solution of the viscous equation is estimated by the method of energy estimates, and the error estimate of H1 is obtained by using Sobolev embedding to obtain L∞ estimate, thus proving the asymptotic equivalence of the two kinds of equations.