Nonclassical Entropy Solutions of the Cauchy Problem

Abstract

In this chapter we give a general existence result for nonclassical entropy solutions to the Cauchy problem associated with a system of conservation laws whose characteristic fields are genuinely nonlinear or concave-convex. (The result can be extended to linearly degenerate and convex-concave fields as well.) The proof is based on a generalization of the algorithm described in Chapter VII. Here, we use the nonclassical Riemann solver based on a given kinetic function for each concave-convex as was described in Section VI-3. Motivated by the examples arising in the applications (see Chapter III) we can assume that the kinetic functions satisfy the following threshold condition: any shock wave with strength less than some critical value is classical. In Section 1 we introduce a generalized total variation functional which is non-increasing for nonclassical solutions (Theorem 1.4) and whose decay rate can be estimated (Theorem 1.5). In Section 2 we introduce a generalized interaction potential and we extend Theorem IV-4.3 to nonclassical solutions; see Theorem 2.1. Section 3 and 4 are concerned with the existence and regularity theory for systems; see Theorems 3.1 and 4.2 respectively.