Viscosity solutions and standard Riemann semigroup for conservation laws with boundary
type: Item
instance of: scholarly article
title: Viscosity solutions and standard Riemann semigroup for conservation laws with boundary
zbMATH Open document ID: 0910.35078
author: Debora Amadori
author: R. M. Colombo
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1999-04-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108474
review text: The authors consider two different formulations of the initial boundary value problem (IBVP) refered as characteristic and noncharacteristic for the following conservation law system: \(u_t+ [F(u)]_x =0\), where \(F\) is smooth and each characteristic field in \(DF\) is either linearly degenerate or genuinely nonlinear. Two ``standard Riemann semigroups'' (SRS) generated by the problem under consideration are defined here. It is shown that if an SRS exists, then it is unique, and its trajectories yield solutions to the IBVP in each of the following two cases: characteristic and noncharacteristic IBVP. Moreover, the trajectories are viscosity solutions of the IBVP for the considered system. Conversely, a viscosity solution with small total variation coincides with the corresponding semigroup trajectory as soon as SRS exists.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L50
zbMATH DE Number: 1183140
zbMATH Keywords: characteristic and noncharacteristic formulation
zbMATH Keywords: viscosity solution with small total variation
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
cites work: Initial-boundary value problems for nonlinear systems of conservation laws
cites work: Continuous dependence for \(2 \times 2\) conservation laws with boundary
cites work: Global solutions of systems of conservation laws by wave-front tracking
cites work: The unique limit of the Glimm scheme
cites work: Q5284536
cites work: The semigroup generated by \(2 \times 2\) conservation laws
cites work: Unique solutions of 2 X 2 conservation laws with large data
cites work: Well-posedness of the Cauchy problem for 𝑛×𝑛 systems of conservation laws
cites work: Boundary conditions for nonlinear hyperbolic systems of conservation laws
cites work: Solutions in the large for nonlinear hyperbolic systems of equations
cites work: Hyperbolic systems of conservation laws II
cites work: Glimm's method and initial- boundary value problem
cites work: Q4314474
Recommended article: Viscous limits for a Riemannian problem to a class of systems of conservation laws
Recommended article: Self‐similar viscosity approach to the Riemann problem for a strictly hyperbolic system of conservation laws
Recommended article: Vanishing viscosity solutions of hyperbolic systems of conservation laws with boundary
Recommended article: Q4942195
Recommended article: Q4376418
Recommended article: Q3483714
Recommended article: Viscosity solutions of second order partial differential equations with boundary conditions on Riemannian manifolds
Recommended article: Q4241922
Recommended article: Viscous limits for piecewise smooth solutions to systems of conservation laws
Recommended article: Q4276410
description: scientific article; zbMATH DE number 1183140