The Riemann problem and the existence of weak solutions to a system of mixed-type in dynamic phase transition
type: Item
instance of: scholarly article
title: The Riemann problem and the existence of weak solutions to a system of mixed-type in dynamic phase transition
zbMATH Open document ID: 0929.35104
author: Harumi Hattori
published in: Journal of Differential Equations
publication date: 1999-07-28T00:00:00Z
review text: The Riemann problem and the existence of weak solutions to a hyperbolic-elliptic mixed system describing a dynamic phase transition problem are discussed. The system is given by \[ v_t- u_x= 0,\quad u_t- f(v)_x= 0, \] where \(v\), \(u\), and \(f\) are strain, velocity, and stress, respectively. To estimate the wave strength of outgoing waves containing the phase boundary, the reflection and transmission coefficients for the phase boundary are obtained. Employing the entropy rate admissibility criterion and the initiation criterion, the existence of global weak solutions is proved via Glimm scheme.
Mathematics Subject Classification ID: 35Q15
Mathematics Subject Classification ID: 35M10
Mathematics Subject Classification ID: 82B26
Mathematics Subject Classification ID: 80A22
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 1212363
zbMATH Keywords: mixed type equation
zbMATH Keywords: dynamic phase transition
zbMATH Keywords: entropy rate
zbMATH Keywords: Glimm scheme
reviewed by: G. Jaiani
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3433
OpenAlex ID: W1984945767
cites work: Kinetic relations and the propagation of phase boundaries in solids
cites work: Stability theorem and truncation error analysis for the Glimm scheme and for a front tracking method for flows with strong discontinuities
cites work: The entropy rate admissibility criterion for solutions of hyperbolic conservation laws
cites work: Q4083057
cites work: A vanishing viscosity approach on the dynamics of phase transitions in van der Waals fluids
cites work: The Uniqueness and Stability of the Solution of the Riemann Problem of a System of Conservation Laws of Mixed Type
cites work: Q4281282
cites work: Solutions in the large for nonlinear hyperbolic systems of equations
cites work: Decay of solutions of systems of nonlinear hyperbolic conservation laws
cites work: The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion-isothermal case
cites work: The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion: Nonisothermal case
cites work: The Entropy Rate Admissibility Criterion and the Entropy Condition for a Phase Transition Problem: The Isothermal Case
cites work: A dynamical system approach to a phase transition problem
cites work: Stability of Coexisting Phases for Compressible Van Der Waals Fluids
cites work: Uniqueness of admissible solutions of the Riemann problem for a system of conservation laws of mixed type
cites work: The propagation of phase boundaries in elastic bars
cites work: Q3732164
cites work: Propagating phase boundaries: Formulation of the problem and existence via the Glimm method
cites work: Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and stability
cites work: Instabilities in Glimm’s Scheme for Two Systems of Mixed Type
cites work: The Riemann problem for a class of conservation laws of mixed type
cites work: Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type
cites work: Admissibility criteria for propagating phase boundaries in a van der Waals fluid
cites work: Dynamic phase transitions in a van der Waals fluid
cites work: A limiting ``viscosity'' approach to the Riemann problem for materials exhibiting change of phase
DOI: 10.1006/JDEQ.1998.3433
Recommended article: Q4247547
Recommended article: Q3480556
Recommended article: On the Riemann problem for some discontinuous systems of conservation laws describing phase transitions
Recommended article: Q4281282
Recommended article: The Existence and Large Time Behavior of Solutions to a System Related to a Phase Transition Problem
Recommended article: The Riemann problem for weakly perturbed \(2\times 2\) hyperbolic systems
Recommended article: The Riemann problem for a system of conservation laws of mixed type with a cubic nonlinearity
Recommended article: Q4376418
Recommended article: Q4954896
Recommended article: An existence theorem and weak Lagrange equations for a variational problem of the theory of phase transitions
description: scientific article; zbMATH DE number 1212363