Shock Waves and Entropy

Abstract

This chapter provides an overview of shock waves and entropy. It describes systems of the first order partial differential equations in conservation form: ∂tU + ∂XF = 0, F = F(u). In many cases, all smooth solutions of the first order partial differential equations in conservation form satisfy an additional conservation law where U is a convex function of u. The chapter discusses that for all weak solutions of ∂tuj +∂xfj= 0, j=1,…, m, fj=fj(u1,…, um), which are limits of solutions of modifications ∂tuj +∂xfj= 0, j=1,…, m, fj=fj(u1,…, um) , by the introduction of various kinds of dissipation, satisfy the entropy inequality, that is, ∂tU + ∂xF≦ 0. The chapter also explains that for weak solutions, which contain discontinuities of moderate strength, ∂tU + ∂xF≦ 0 is equivalent to the usual shock condition involving the number of characteristics impinging on the shock. The chapter also describes all possible entropy conditions of ∂tU + ∂xF≦ 0 that can be associated to a given hyperbolic system of two conservation laws.