Regularity Through Approximation for Scalar Conservation Laws

In this paper it is shown that recent approximation results for scalar conservation laws in one space dimension imply that solutions of these equations with smooth, convex fluxes have more regularity than previously believed. Regularity is measured in spaces determined by quasinorms related to the solution's approximation properties in L1(R) by discontinuous, piecewise linear functions. Using a previous characterization of these approximation spaces in terms of Besov spaces, it is shown that there is a one-parameter family of Besov spaces that are invariant under the differential equation. An intriguing feature of this investigation is that regularity is measured quite naturally in smoothness classes that are not locally convex—they are similar to Lp spaces for 0 < p < 1. Extensions to Hamilton-Jacobi equations are mentioned.