High Order Regularity for Conservation Laws

We study the regularity of discontinuous entropy solutions to scalar hyperbolic conservation laws with uniformly convex fluxes posed as initial value problems on R. For positive α we show that if the initial data has bounded variation and the flux is smooth enough then the solution u( · , t) is in the Besov space Bα σ (L σ) where σ = 1/(α + 1) whenever the initial data is in this space. As a corollary, we show that discontinuous solutions of conservation laws have enough regularity to be approximated well by moving-grid finite element methods. Techniques from approximation theory are the basis for our analysis.