Hugoniot Relation for Multi-Temperature Euler Equations of Compressible Plasma Flows

TL;DR

This paper analyzes shock solutions in multi-temperature Euler equations for plasma flows, revealing non-uniqueness in Hugoniot relations and emphasizing the importance of microscopic physics in resolving shock ambiguities.

Contribution

It derives two physically admissible Hugoniot relations for multi-temperature plasma flows, highlighting the non-uniqueness and the need for external physical input.

Findings

Two distinct Hugoniot relations are derived for multi-temperature shocks.

Both relations satisfy classical admissibility conditions, indicating shock non-uniqueness.

The Hugoniot relation must be informed by experiments or microscopic physics, not just PDEs.

Abstract

Shock solutions for multi-temperature Euler equations are inherently ambiguous due to the loss of microscopic physical detail during model reduction and occurrence of non-conservative terms. This paper presents a detailed analytical study of shock structures in such models. We derive two distinct Hugoniot relations, each corresponding to a physically admissible shock solution: one for the general multi-temperature case and one for two-temperature plasma flows. Through classical analysis \`a la Courant--Friedrichs, we demonstrate that both satisfy admissibility conditions, revealing a fundamental non-uniqueness in shock structures. By relating these solutions to existing numerical schemes, the structure preserving and vanishing viscosity approaches, we provide physically justified references for constructing and evaluating discontinuous numerical approximations. In particular, we emphasize that the Hugoniot relation is not uniquely determined by the macroscopic PDEs alone, but must be supplied from external sources such as experiments or first-principles simulations. This insight demonstrates the essential role of microscopic physics in resolving shock ambiguity and contributes to the theoretical foundation for modeling discontinuous plasma flows.