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

Shih-Wei Chou (Department of Finance Engineering and Actuarial Mathematics, Soochow University, Taipei, Taiwan)

John M. Hong (Department of Mathematics, National Central University, Jhongli City, Taiwan)

Ying-Chieh Lin (Department of Applied Mathematics, National University of Kaohsiung, Taiwan)

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

Full Text (PDF format)

Received 18 March 2019

Accepted 23 August 2019

Published 13 May 2020