Diffusive–Dispersive Traveling Waves and Kinetic Relations: Part I: Nonconvex Hyperbolic Conservation Laws

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, uniquen...

Full description

Saved in:
Bibliographic Details
Published in:Journal of Differential Equations 2002-01, Vol.178 (2), p.574-607
Main Authors: Bedjaoui, Nabil, LeFloch, Philippe G.
Format: Article
Language:English
Subjects:
diffusion
dispersion
entropy inequality
hyperbolic conservation law
kinetic relation
shock wave
undercompressive
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0022-0396
1090-2732
DOI:10.1006/jdeq.2000.4009