Dispersive regularization for phase transitions

TL;DR

This paper introduces a dispersive regularization method for the one-dimensional compressible Euler equations with Van der Waals pressure law, capturing complex behaviors near phase transitions.

Contribution

It proposes a novel Schroedinger-type dispersive regularization for Euler equations with phase transition modeling in Lagrangian coordinates.

Findings

Supports high-frequency solutions with variable existence times

Conservation law aligns with physical energy for real solutions

Handles hyperbolic and elliptic zones in pressure law

Abstract

We introduce a dispersive regularization of the compressible Euler equations in Lagrangian coordinates, in the one-dimensional torus. We assume a Van der Waals pressure law, which presents both hyperbolic and elliptic zones. The dispersive regularization is of Schroedinger type. In particular, the regularized system is complex-valued. It has a conservation law, which, for real unknowns, is identical to the energy of the unregularized physical system. The regularized system supports high-frequency solutions, with an existence time or an amplitude which depend strongly on the pressure law.