Delta shock wave for ternary nonlinear chromatography equations as self-similar viscosity limit

Abstract

In this paper, we investigate the existence and stability of the delta shock wave in ternary chromatography equations by the self-similar viscosity vanishing approach. Considering the appropriate initial values, we prove the existence of the self-similar solution for the corresponding Riemann problem of the ternary chromatography viscous equations. Furthermore, we rigorously demonstrate that the delta shock wave is the weak star limit of the self-similar solution as viscosity tends to disappear. The result implies that the structure of the delta shock wave is stable under the self-similar viscosity perturbation, which guarantees that the delta shock wave is a unique entropy solution. In addition, we present numerical simulations in agreement with the theoretical analysis.

Keywords:

• Ternary chromatography equations
• self-similar viscosity vanishing approach
• delta shock wave
• numerical simulation

2020 Mathematics Subject Classifications:

• 35D40
• 35D30

Acknowledgments

The authors thank the editors and the reviewers for their useful feedback that improved this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partially supported by National Natural Science Foundation of China (12161084), the Natural Science Foundation of Xinjiang, PR China (2022D01E42).