Local error estimates for discontinuous solutions of nonlinear hyperbolic equations

Let $u(x,t)$ be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose $u_\varepsilon (x,t)$ is the solution of an approximate viscosity regularization, where $\varepsilon > 0$ is the small viscosity amplitude. It is shown that by post-processing the small viscosity approximation $u_\varepsilon $, pointwise values of u and its derivatives with an error as close to e as desired can be recovered.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 coefficients. The novelty of our approach is to use a (generalized) E-condition of the forward problem in order to deduce a $W^{1,\infty } $-energy estimate for the discontinuous backward transport equation; this, in turn, leads to e-uniform estimate on moments of the error $u_\varepsilon - u$.The approach presented does not “follow the characteristics” and, therefore, applies mutatis mutandis to o...