Zero Mach number limit of the compressible Euler\u2013Korteweg equations

Yeping Li,Gang Zhou

doi:10.1186/s13661-020-01395-4

Abstract

In this paper, we investigate the zero Mach number limit for the three-dimensional compressible Euler–Korteweg equations in the regime of smooth solutions. Based on the local existence theory of the compressible Euler–Korteweg equations, we establish a convergence-stability principle. Then we show that when the Mach number is sufficiently small, the initial-value problem of the compressible Euler–Korteweg equations has a unique smooth solution in the time interval where the corresponding incompressible Euler equations have a smooth solution. It is important to remark that when the incompressible Euler equations have a global smooth solution, the existence time of the solution for the compressible Euler–Korteweg equations tends to infinity as the Mach number goes to zero. Moreover, we obtain the convergence of smooth solutions for the compressible Euler–Korteweg equations towards those for the incompressible Euler equations with a convergence rate.

Highlights

We are concerned with the three-dimensional compressible Euler–Korteweg system
The unknown functions are the density ρ and the velocity u ∈ R3, p(ρ) is a given pressure function, and κ is the Weber number. This compressible Euler–Korteweg system results from a modification of the standard Euler equations governing the motion of compressible inviscid fluids through the adjunction of the Korteweg stress tensor, and arises as a mathematical model for a lot of phenomena in vortex dynamics, quantum hydrodynamics and hydrodynamics, e.g., flow of capillary fluids: liquid-vapor mixtures, superfluids
This completes the proof of Theorem 4.1

Summary

Introduction

We are concerned with the three-dimensional compressible Euler–Korteweg system. We analyze the incompressible limit of smooth solutions for the compressible Euler–Korteweg equations (1.1) with well-prepared initial data on the basis of the convergence-stability criterion, which was first formulated in [37]. Since this method has commonly been used in dealing with the singular limit of the partial differential equations. Remark 2.2 Here, we only consider the zero-Mach limit of the smooth solutions for the compressible Euler–Korteweg equations with well-prepared initial data.

Note that

Uα dx

Then we have

Recall that