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# Methods and Applications of Analysis

## Volume 26 (2019)

### Number 4

### Global existence and strong trace property of entropy solutions by the source-concentration Glimm scheme for nonlinear hyperbolic balance laws

Pages: 371 – 394

DOI: https://dx.doi.org/10.4310/MAA.2019.v26.n4.a4

#### Authors

#### Abstract

In this paper, we investigate the initial-boundary value problem for a nonlinear hyperbolic system of balance laws with source terms $a_x g$ and $a_t h$. We assume that the boundary data satisfy a linear or smooth nonlinear relation. The generalized Riemann and boundary Riemann solutions are provided with the variation of a concentrated on a thin T-shaped region in each grid. We generalize Goodman’s boundary interaction estimates [7], introduce a new version of Glimm scheme to construct the approximation solutions, and provide their stability by considering two types of functions of $a(x, t)$. The global existence of entropy solutions is established. Under some sampling condition, we find the entropy solutions converge to their boundary values in $L^1_{\mathrm{loc}}$ as $x$ approaches the boundary. In addition, such boundary values match the boundary condition almost everywhere in $t$.

#### Keywords

nonlinear balance laws, initial-boundary value problem, Riemann problem, generalized Glimm scheme, concentration of source, wave interaction estimates, entropy solutions, boundary regularity

#### 2010 Mathematics Subject Classification

35L60, 35L65, 35L67

Received 18 March 2019

Accepted 23 August 2019

Published 13 May 2020