An error estimate for viscous approximate solutions of degenerate parabolic equations

Abstract

Relying on recent advances in the theory of entropy solutions for nonlinear (strongly) degenerate parabolic equations, we present a direct proof of an L^1 error estimate for viscous approximate solutions of the initial value problem for \partial_t w+\mathrm{div} \bigl(V(x)f(w)\bigr)= \Delta A(w) where V=V(x) is a vector field, f=f(u) is a scalar function, and A'(.) \geq 0. The viscous approximate solutions are weak solutions of the initial value problem for the uniformly parabolic equation \partial_t w^{\epsilon}+\mathrm{div} \bigl(V(x) f(w^{\epsilon})\bigr) \Delta \bigl(A(w^{\epsilon})+\epsilon w^{\epsilon}\bigr), \epsilon>0. The error estimate is of order \sqrt{\epsilon}.