Viscosity approximation of the solution to Burgers' equations with shock layers

Abstract

Viscous Burgers' equations with a small viscosity are considered and convergence of vanishing viscosity limit problem is investigated. We examine interior layers of a solution to viscous Burgers' equations, u(epsilon), as a viscosity parameter epsilon tends to zero. The inviscid model, i.e. when epsilon = 0, possesses the structure of scalar hyperbolic conservation laws, hence our studies deliver an important idea that arises in the field of shock discontinuities of nonlinear hyperbolic waves. The heart of the paper is to establish asymptotic expansions and utilize inner solutions of sharp transition, which are called a corrector function. With aid of corrector functions and energy estimates, we improve the convergence rate of ue to u(0) as O(epsilon(1/2)) in L-2(R) (O(epsilon) in L-loc(1)(R)) in the regions including shocks under an entropy condition.