Local Error Estimates for Discontinuous Solutionsof Nonlinear Hyperbolic

Let u(x; t) be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose u"(x; t) is the solution of an approximate viscosity regularization, where " > 0 is the small viscosity amplitude. We show that by post-processing the small viscosity approximation u", we can recover pointwise values of u and its derivatives with an error as close to " as desired. The analysis relies on the adjoint problem of the forward error equation, which in this case amounts to a backward linear transport equation with discontinuous coeecients. The novelty of our approach is to use a (generalized) E-condition of the forward problem in order to deduce a W 1;1-energy estimate for the discontinuous backward transport equation; this, in turn, leads us to "-uniform estimate on moments of the error u" ? u. Our approach does not`follow the characteristics' and, therefore, applies mutatis mutandis to other approximate solutions such as E-diierence schemes.