High-friction limits of Euler flows for multicomponent systems

TL;DR

This paper investigates the high-friction limit of Euler-Korteweg equations for fluid mixtures, establishing convergence to limiting systems and characterizing the first-order correction as Maxwell-Stefan type with parabolic diffusive behavior.

Contribution

It proves the convergence towards zeroth- and first-order systems in high-friction regimes, identifying the first-order correction as a Maxwell-Stefan type system with parabolic diffusive properties.

Findings

First-order correction system is Maxwell-Stefan type.

Diffusive part of the correction is parabolic.

Convergence established for both smooth and weak solutions.

Abstract

The high-friction limit in Euler-Korteweg equations for fluid mixtures is analyzed. The convergence of the solutions towards the zeroth-order limiting system and the first-order correction is shown, assuming suitable uniform bounds. Three results are proved: The first-order correction system is shown to be of Maxwell-Stefan type and its diffusive part is parabolic in the sense of Petrovskii. The high-friction limit towards the first-order Chapman-Enskog approximate system is proved in the weak-strong solution context for general Euler-Korteweg systems. Finally, the limit towards the zeroth-order system is shown for smooth solutions in the isentropic case and for weak-strong solutions in the Euler-Korteweg case. These results include the case of constant capillarities and multicomponent quantum hydrodynamic models.