Stability properties of the Euler–Korteweg system with nonmonotone pressures

Abstract

We establish a relative energy framework for the Euler–Korteweg system with non-convex energy. This allows us to prove weak-strong uniqueness and to show convergence to a Cahn–Hilliard system in the large friction limit. We also use relative energy to show that solutions of Euler–Korteweg with convex energy converge to solutions of the Euler system in the vanishing capillarity limit, as long as the latter admits sufficiently regular strong solutions.

Keywords:

• Euler–Kortweg system
• phase transitions
• Cahn–Hiliard equation

AMS Subject Classifications:

• 35Q31
• 76D45
• 76T10

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

JG thanks the Baden-Wuerttemberg foundation for support via the project `Numerical Methods for Multiphase Flows with Strongly Varying Mach Numbers‘.