Compensated Compactness

Abstract

Approximate solutions to hyperbolic systems of conservation laws may be generated in a variety of ways: by the method of vanishing viscosity, through difference approximations, by relaxation schemes, etc. The topic for discussion in this chapter is whether solutions may be constructed as limits of sequences of approximate solutions that are bounded only in some L ^p space. Since the systems are nonlinear, the difficulty lies in that the construction schemes are generally consistent only when the sequence of approximating solutions converges strongly, whereas the assumed L ^p bounds guarantee only weak convergence: Approximate solutions may develop high-frequency oscillations of finite amplitude which play havoc with consistency.The aim is to demonstrate that entropy inequalities may save the day by quenching rapid oscillations, thus enforcing strong convergence of the approximating solutions. Some indication of this effect was alluded to in Section 1.9. The principal tools in the investigation will be the notion of Young measure and the functional analytic method of compensated compactness. The former naturally induces the very general class of measure-valued solutions and the latter is employed to verify that nonlinearity reduces measure-valued solutions to traditional ones. As it relies heavily on entropy dissipation, the approach appears to be applicable mainly to systems endowed with a rich family of entropy-entropy flux pairs, most notably the scalar conservation law and systems of just two conservation laws. Despite this limitation, the approach is quite fruitful, not only because of the abundance of important systems with such structure, but also because it provides valuable insight into the stabilizing role of entropy dissipation as well as into the “conflicted” stabilizingdestabilizing behavior of nonlinearity. Different manifestations of these factors were already encountered in earlier chapters.