Shock Waves, Increase of Entropy and Loss of Information

摘要

We discuss, for the simplified model of a single conservation law, the concepts of genuine nonlinearity, breakdown of classical solutions, solutions in the distribution sense and their nonuniqueness, the viscosity method, finite difference methods, and the shock condition. We then discuss, for the scalar model, the compactness of solutions constructed by the viscosity and difference methods, and derive the entropy inequality for such solutions. We derive Glimm's estimate for the total variation of solutions of scalar equations that satisfy the shock condition, and show that a discontinuous solution that satisfies the shock condition also satisfies the entropy condition. Scattered remarks are given about the equations of compressible flow: the increase of entropy, some consequences of Carnot's theorem, and the equipartition of energy in the wake of strong shocks. (ERA citation 10:038028)