Contents Online

# Methods and Applications of Analysis

## Volume 19 (2012)

### Number 1

### Global entropy solutions of the general nonlinear hyperbolic balance laws with time-evolution flux and source

Pages: 43 – 76

DOI: http://dx.doi.org/10.4310/MAA.2012.v19.n1.a3

#### Authors

#### Abstract

In this paper we investigate the initial and initial-boundary value problems for strictly hyperbolic balance laws with time-evolution of flux and source. Such nonlinear balance laws arise in, for instance, gas dynamics equations in time-dependent ducts and nozzles, shallow water equations, lanes-changing model in traffic flow and Einstein’s field equations in a spherically symmetric spacetime. To account for the time dependence of flux and source, we introduce the perturbed Riemann and boundary Riemann problems. Such Riemann problems have unique solutions within elementary waves and an additional family of waves. Based on the work of [12, 13], a new version of Glimm scheme is introduced and its stability is established by modified interaction estimates. Finally, the existence of global entropy solutions is achieved by showing the consistency of scheme, the weak convergence of source term and the entropy inequalities.

#### Keywords

initial value problem, initial-boundary value problem, hyperbolic conservation laws, nonlinear balance laws, entropy solutions, Riemann problem, perturbed Riemann problem, perturbed boundary Riemann problem, wave interaction estimates, generalized Glimm scheme

#### 2010 Mathematics Subject Classification

35L60, 35L65, 35L67