Extending relative entropy for Korteweg-Type models with non-monotone pressure:large friction limit and weak-strong uniqueness

TL;DR

This paper extends the relative entropy method to analyze weak-strong uniqueness and relaxation limits for Korteweg-type fluid models with non-monotone pressure, generalizing previous results to more complex capillarity functions.

Contribution

It introduces an enlarged formulation with drift velocity to handle non-monotone pressure, broadening the applicability of relative entropy techniques to these models.

Findings

Proves weak-strong uniqueness for the models.

Establishes singular relaxation limits.

Generalizes previous results to non-monotone pressure cases.

Abstract

In this paper we study weak-strong uniqueness and singular relaxation limits for the Euler--Korteweg and Navier--Stokes--Korteweg systems with non monotone pressure. Both weak-strong uniqueness and the relaxation limit are investigated using relative entropy technique. We make use of the enlarged formulation of the model in terms of the drift velocity introduced in [6], generalizing in this way results proved in [17] for the Euler-Korteweg model, by allowing more general capillarity functions, and the result contained in [8] for the monotone pressure case.