Global Weak Solutions for a Nonlinear Hyperbolic System

Abstract

In this paper, we study the global existence of weak solutions for the Cauchy problem of the nonlinear hyperbolic system of three equations (1.1) with bounded initial data (1.2). When we fix the third variable s, the system about the variables ρ and u is the classical isentropic gas dynamics in Eulerian coordinates with the pressure function $P(\rho ,s) = {{\rm{e}}^s}{{\rm{e}}^{ - {1 \over \rho }}}$, which, in general, does not form a bounded invariant region. We introduce a variant of the viscosity argument, and construct the approximate solutions of (1.1) and (1.2) by adding the artificial viscosity to the Riemann invariants system (2.1). When the amplitude of the first two Riemann invariants (w₁(x, 0), w₂(x, 0)) of system (1.1) is small, (w₁(x, 0), w₂(x, 0)) are nondecreasing and the third Riemann invariant s(x, 0) is of the bounded total variation, we obtained the necessary estimates and the pointwise convergence of the viscosity solutions by the compensated compactness theory. This is an extension of the results in [1].