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

• Published in: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2002-06, Vol.132 (3), p.545-565
• Main Authors: Bedjaoui, Nabil, LeFloch, Philippe G.
• Format: Article
• Language: English
• ISSN: 0308-2105; 1473-7124
• DOI: 10.1017/S0308210500001773

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.