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)= ΔA(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^ε+\mathrm{div} \bigl(V(x) f(w^ε)\bigr) Δ\bigl(A(w^ε)+εw^ε\bigr), ε>0. The error estimate is of order \sqrtε.

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