Convergence of viscosity solutions for \(2\times 2\) hyperbolic conservation laws with one characteristic field linearly degenerate on some zero measure sets
type: Item
instance of: scholarly article
title: Convergence of viscosity solutions for \(2\times 2\) hyperbolic conservation laws with one characteristic field linearly degenerate on some zero measure sets
zbMATH Open document ID: 0851.35091
author: Changjiang Zhu
published in: Chinese Science Bulletin
publication date: 1996-11-25T00:00:00Z
review text: Consider the Cauchy problem for the general \(2\times 2\) hyperbolic conservation laws \[ u_t+ f(u, v)_x= 0,\quad v_t+ g(u, v)_x= 0,\tag{1} \] \[ (u, v)(x, 0)= (u_0(x), v_0(x)).\tag{2} \] Suppose that the two eigenvalues of system (1) are \(\lambda_1(u, v)\), \(\lambda_2(u, v)\), the corresponding Riemann invariants are \(w= w(u, t)\), \(z= z(u, v)\), and \(w= w(u, v)\), \(z= z(u, v)\) give a bijective smooth mapping from \((u, v)\) plane onto \((w, z)\) plane. We suppose that \(u_0(x)\), \(v_0(x)\) are bounded measurable functions, \(\lambda_1(u, v)< \lambda_2(u, v)\) and that one characteristic field of system (1) is genuinely nonlinear, and the other is linearly degenerate on some zero measure sets. In this note, we prove the convergence of viscosity solutions through analyzing the progressing entropy waves of Lax type, and we establish an existence theorem of the global weak solution for the Cauchy problem (1), (2).
Mathematics Subject Classification ID: 35L65
zbMATH DE Number: 883561
zbMATH Keywords: viscosity solutions
zbMATH Keywords: global weak solution
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 883561