Existence of traveling waves associated with Lax shocks which violate Oleinik’s entropy criterion

Abstract

This paper answers to the question whether a shock wave in conservation laws satisfying the Lax shock inequalities but not Oleinik’s entropy criterion is admissible under the vanishing viscosity-capillarity effects. Such a shock appears in van der Waals fluids when a secant line meets the graph of the flux function at four distinct points, and the shock jumps between the two farthest points. The existence of the corresponding traveling waves would justify the admissibility of the shock. For this purpose, we will first show that the corresponding traveling waves satisfy a system of differential equations with two saddle points and two asymptotically stable points. Second, we estimate the domains of attraction of the asymptotically stable equilibrium points, relying on Lyapunov’s stability theory. Third, we investigate the circumstances when an unstable trajectory leaving the saddle point corresponding to the left-hand state of the shock will ever enter the domain of attraction of each of the two asymptotically stable equilibrium points. Finally, we establish the existence of traveling waves associated with a Lax shock but violating the Oleinik’s entropy criterion.

Keywords:

• Shock wave
• traveling wave
• Lax shock inequalities
• Oleinik’s entropy criterion
• viscosity
• capillarity
• Lyapunov stability

AMS Subject Classifications:

• 35L65
• 74N20
• 76N10
• 76L05

Acknowledgements

The authors would like to thank the reviewers for their very constructive comments and fruitful discussions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) [grant number B2015-28-02].