Diffusive–Dispersive Traveling Waves and Kinetic Relations

Abstract

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

Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

“…Nonclassical solutions have the distinctive feature of being dynamically driven by small-scale effects such as diffusion, dispersion, and other high-order phenomena. Their selection requires an additional jump relation, called a kinetic relation, and introduced in the context of phase transition dynamics by Slemrod [35,36,13], Truskinovsky [37,38], Abeyaratne and Knowles [1,2], LeFloch [23], and Shearer [33,34], and developed in the more general context of nonlinear hyperbolic systems of conservation laws by LeFloch and collaborators [15]- [17], [3]- [5], [28]- [30], and [27]. See [24] for a review.…”

“…Note in passing that this solver was later extended in [27] to include also a nucleation criterion. One considers the problem (1)- (2)- (5) for a given Riemann initial data (3). Throughout this paper we assume that the flux f is either concave-convex or convex-concave, that is, satisfies the conditions…”

Travelling wave solutions for the Richards equation incorporating non-equilibrium effects in the capillarity pressure

“…A similar analysis, but for two-phase flow models implying convex-concave flux functions is carried out in [44,47,48]. Also related are the diffusive-dispersive equations appearing as models for the phase transition dynamics, but in which the higher order terms are in terms of the spatial derivatives only [2,16]. Though having a different motivation, the associated TW equation is similar to the one for the dynamic capillarity models, in particular since both involve a non-convex nonlinearity in the lower order terms.…”

Zero Diffusion-Dispersion Limits for Scalar Conservation Laws

“…holds, we prove that the limit coincides with the entropy solution determined by Kruzkov's theory [10]. We point out that these conditions are sharp since, in the limiting case, δ = K ǫ 2 for some K ∈ R I , (1.5) limiting solutions may violate Kruzkov's entropy conditions [8,5,13,2]. Furthermore, when (1.3) is violated, the solutions are highly-oscillatory and fail to converge in any strong topology as noted by Lax and Levermore [12].…”