FORMATION OF DIFFUSION WAVES IN A SCALAR CONSERVATION LAW WITH CONVECTION

We study the scalar conservation law, u(t)+[c(x)u]x+b(u(2))x = u(xx), c' greater than or equal to 0, which is a model equation for the behavior of weak transverse waves near a viscous transitional shock. Solutions are shown to decay in L(1) to a pair of diffusion waves, moving apart at speeds c(-infinity) and c(+infinity), behavior that has been observed numerically in solutions of the full equations. The interesting aspect of the analysis is that the asymptotic state of the solution is not known a priori, in contrast to cases treated previously.