Existence of traveling waves to any Lax shock satisfying Oleinik’s criterion in conservation laws

Abstract

Given any shock wave of a conservation law where the flux function may not be convex, we want to know whether it is admissible under the criterion of vanishing viscosity/capillarity effects. In this work, we show that if the shock satisfies the Oleinik’s criterion and the Lax shock inequalities, then for an arbitrary diffusion coefficient, we can always find suitable dispersion coefficients such that the diffusive-dispersive model admits traveling waves approximating the given shock. The paper develops the method of estimating attraction domain for traveling waves we have studied before.

Keywords:

• Conservation laws
• traveling wave
• shock
• diffusion
• dispersion
• equilibria
• asymptotical stability
• Lyapunov function
• LaSalle’s invariance principle
• attraction domain

AMS Subject Classifications:

• 35L65
• 74N20
• 76N10
• 76L05