Motivated by the theory of phase transition dynamics, we consider one-dimensional, nonlinear hyperbolic conservation laws with nonconvex flux-function containing vanishing nonlinear diffusive-dispersive terms. Searching for traveling wave solutions, we establish general results of existence, uniqueness, monotonicity, and asymptotic behavior. In particular, we investigate the properties of the traveling waves in the limits of dominant diffusion, dominant dispersion, and asymptotically small or large shock strength. As the diffusion and dispersion parameters tend to 0, the traveling waves converge to shock wave solutions of the conservation law, which either satisfy the classical Oleinik entropy criterion or are nonclassical undercompressive shocks violating it. © 2002 Elsevier Science

We consider scalar hyperbolic conservation laws with non-convex flux and vanishing, nonlinear and possibly singular, diffusion and dispersion terms. The diffusion has the form R (u, ux)x and we cover, for instance, the singular diffusion (|ux| p ux)x, where p 0 is arbitrary. We investigate the existence, uniqueness and various properties of classical and non-classical travelling waves and of the kinetic function. The latter serves to characterize non-classical shock waves, via an additional algebraic constrain called a kinetic relation. We discover that p = 1 3 is a somewhat unexpected critical value. For p 1 3 , we obtain properties that are qualitatively similar to those we established earlier for regular and linear diffusion. However, for p > 1 3 , the behaviour of the kinetic function is very different, as, for instance, non-classical shocks can have arbitrary small strength. The behaviour of the kinetic function near the origin is carefully investigated and depends on whether p < 1 2 , p = 1 2 or p > 1 2 . In particular, in the special case of the cubic flux-function and for the regularization (|ux| p ux)x with p = 0, 1 2 or 1, the kinetic function can be computed explicitly. When p = 1 2 , the kinetic function is simply a linear function of its argument.

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.