On the oscillatory solutions in hyperbolic conservation laws
type: Item
instance of: scholarly article
title: On the oscillatory solutions in hyperbolic conservation laws
zbMATH Open document ID: 0979.35097
DOI: 10.1016/S0362-546X(99)00207-2
author: George R. Peters
author: Sunčica Čanić
published in: Nonlinear Analysis. Real World Applications
publication date: 2002-02-21T00:00:00Z
review text: The authors study Riemann problems with shock initial data for two examples of \(2\times 2\) conservation laws in one-space dimension: the shallow water equations and a three-phase Buckley-Leverett flow. For shock initial data that do not possess a viscous profile, they consider a parabolic regularization of the conservation laws for which they construct approximate oscillatory solutions by the Crank-Nicholson method. Numerical examples convince them that the approximate solutions are uniformly bounded; admitting this, they prove convergence of subsequences to measure-valued solutions of the system of conservation laws with the given initial data.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 76M20
zbMATH DE Number: 1529221
zbMATH Keywords: oscillatory solutions
zbMATH Keywords: Riemann problems
zbMATH Keywords: shock initial data
zbMATH Keywords: viscous profile
zbMATH Keywords: parabolic regularization
zbMATH Keywords: Crank-Nicholson method
zbMATH Keywords: measure-valued solutions
reviewed by: Paul Godin
MaRDI profile type: Publication
cites work: Q4040134
cites work: An implicit finite-difference algorithm for hyperbolic systems in conservation-law form
cites work: Quadratic systems of conservation laws with generic behavior at infinity
cites work: Nonexistence of Riemann solutions and Majda-Pego instability
cites work: Shock wave admissibility for quadratic conservation laws
cites work: Viscosity matrices for two‐dimensional nonlinear hyperbolic systems
cites work: Measure-valued solutions to conservation laws
cites work: Oscillations and concentrations in weak solutions of the incompressible fluid equations
cites work: Steady state solutions of hyperbolic systems of conservation laws with viscosity terms
cites work: Q4887450
cites work: Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type
cites work: Oscillation waves in Riemann problems for phase transitions
cites work: Some problems in the theory of quasilinear equations
cites work: The Riemann Problem Near a Hyperbolic Singularity: The Classification of Solutions of Quadratic Riemann Problems I
cites work: Hyperbolic systems of conservation laws II
cites work: The entropy condition and the admissibility of shocks
cites work: Q4889859
cites work: Stable viscosity matrices for systems of conservation laws
cites work: A topological degree for orbits connecting critical points of autonomous systems
cites work: The classification of 2 × 2 systems of non-strictly hyperbolic conservation laws, with application to oil recovery
cites work: Q4746129
cites work: Q3878033
full work available at URL: s0362-546x(99)00207-2
OpenAlex ID: W2076575476
description: scientific article; zbMATH DE number 1529221