Shock waves, increase of entropy and loss of information

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 are discussed. For the scalar model, the compactness of solutions constructed by the viscosity and difference methods are also discussed, and the entropy inequality for such solutions is derived. Glimm's estimate for the total variation of solutions of scalar equations that satisfy the shock condition is also derived, and It is shown 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.