Admissible Shocks

Abstract

Shock frontswere introduced in Section 1.6, for general systems of balance laws, and were placed in the context of BV solutions in Section 1.8. They were encountered again, briefly, in Section 3.1, where the governing Rankine-Hugoniot condition was recorded. Since shock fronts have codimension one, important aspects of their local behavior may be investigated, without loss of generality, within the framework of systems in one space dimension. This will be the object of the present chapter. The discussion will begin with an exploration of the geometric features of the Rankine-Hugoniot condition, leading to the introduction of the Hugoniot locus. The necessity of imposing admissibility conditions on weak solutionswas pointed out in Chapter IV. These in turn induce, or at least motivate, admissibility conditions on shocks. Indeed, the prevailing view is that the issue of admissibility of general BV weak solutions should be resolved through a test applied to every point of the shock set. In particular, the shock admissibility conditions associated with the entropy condition of Section 4.5 and the vanishing viscosity approach of Section 4.6 will be introduced, and they will be compared with each other as well as with other important shock admissibility conditions proposed by Lax and by Liu.