Diffusive-dispersive travelling waves and kinetic relations. II A hyperbolic–elliptic model of phase-transition dynamics

We deal here with a mixed (hyperbolic-elliptic) system of two conservation laws modelling phase-transition dynamics in solids undergoing phase transformations. These equations include nonlinear viscosity and capillarity terms. We establish general results concerning the existence, uniqueness and asymptotic properties of the corresponding travelling wave solutions. In particular, we determine their behaviour in the limits of dominant diffusion, dominant dispersion or asymptotically small or large shock strength. As the viscosity and capillarity parameters tend to zero, the travelling waves converge to propagating discontinuities, which are either classical shock waves or supersonic phase boundaries satisfying the Lax and Liu entropy criteria, or else are undercompressive subsonic phase boundaries. The latter are uniquely characterized by the so-called kinetic function, whose properties are investigated in detail here.

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Non-classical Riemann solvers and kinetic relations. II. An hyperbolic-elliptic model of phase-transition dynamics

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210502000094 ◽

2002 ◽

Vol 132 (01) ◽

pp. 181

Keyword(s):

Phase Transition ◽

Transition Dynamics ◽

Riemann Solvers ◽

Kinetic Relations ◽

Elliptic Model

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Non-classical Riemann solvers and kinetic relations. II. An hyperbolic–elliptic model of phase-transition dynamics

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821050000158x ◽

2002 ◽

Vol 132 (1) ◽

pp. 181-219 ◽

Cited By ~ 19

Author(s):

Philippe G. LeFloch ◽

Mai Duc Thanh

Keyword(s):

Phase Transition ◽

Riemann Problem ◽

Entropy Inequality ◽

Phase Boundaries ◽

Kinetic Relation ◽

Two Phase ◽

Transition Dynamics ◽

Riemann Solvers ◽

All Solutions ◽

Elliptic Model

This paper deals with the Riemann problem for a partial differential equation's model arising in phase-transition dynamics and consisting of an hyperbolic–elliptic system of two conservation laws. First of all, we provide a complete description of all solutions of the Riemann problem that are consistent with the mathematical entropy inequality associated with the total energy of the system. Second, following Abeyaratne and Knowles, we impose a kinetic relation to determine the dynamics of subsonic phase boundaries. Based on the requirement that subsonic phase boundaries are preferred whenever available, we determine the corresponding wave curves associated with composite waves (shocks, rarefaction fans, phase boundaries). It turns out that even after the kinetic relation is imposed, the Riemann problem may admit up to two solutions. A nucleation criterion is necessary to select between a solution remaining in a single phase and a solution containing two phase boundaries. Alternatively, a strong assumption on the kinetic relation ensures that the Riemann solution is unique and depends continuously upon its initial data.

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GLOBAL EXISTENCE FOR PHASE TRANSITION PROBLEMS VIA A VARIATIONAL SCHEME

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891604000329 ◽

2004 ◽

Vol 01 (04) ◽

pp. 747-768

Author(s):

CHRISTIAN ROHDE ◽

MAI DUC THANH

Keyword(s):

Phase Transition ◽

Surface Tension ◽

Initial Data ◽

Time Discretization ◽

Approximate Solutions ◽

Implicit Time ◽

Time Step ◽

Dynamical Phase Transition ◽

Transition Dynamics ◽

Variational Scheme

We construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension. First, we deal with a local model of phase transition dynamics which contains second and third order spatial derivatives modeling the effects of viscosity and surface tension. Assuming that the initial data are periodic, we prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality. We note that this result still holds for non-periodic initial data. Second, we consider a model of phase transition dynamics with only Lipschitz continuous stress–strain function which contains a non-local convolution term to take account of surface tension. We also establish the existence of weak solutions. In both cases the proof relies on implicit time discretization and the analysis of a minimization problem at each time step.

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