Explicit formula for the limit of a difference approximation

Abstract

We consider the conservation law (1.1) u,+[log (ae’‘+ be-’‘)] x 0 where a> 0, b> 0, a+ b 1, with initial condition (1.2) u (x, O) Uo (X) in the quarter plane x> 0,> 0. From the work of Bardos, Leroux andNedelec [1] and Dubois and Lefloch [2-1, it is known that one cannot prescribe u (0, t) 2 (0 arbitrarily and hope to have a solution for (1.1) and (1.2) which satisfy this boundary condition. In fact they considered parabolic approximations of general scalar con-servation laws and showedthat, as the viscosity goes to zero, the limit of the approximate solution satisfies a boundary entropy inequality at the boundary.