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# Communications in Mathematical Sciences

## Volume 14 (2016)

### Number 1

### Zero Mach number limit of the compressible Navier–Stokes–Korteweg equations

Pages: 233 – 247

DOI: https://dx.doi.org/10.4310/CMS.2016.v14.n1.a9

#### Authors

#### Abstract

In this paper, we investigate the zero Mach number limit for the three-dimensional compressible Navier–Stokes–Korteweg equations in the regime of smooth solutions. Based on the local existence theory of the compressible Navier–Stokes–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 Navier–Stokes–Korteweg equations has a unique smooth solution in the time interval where the corresponding incompressible Navier–Stokes equations have a smooth solution. It is important to remark that when the incompressible Navier–Stokes equations have a global smooth solution, the existence time of the solution for the compressible Navier–Stokes–Korteweg equations tends to infinity as the Mach number goes to zero. Moreover, we obtain the convergence of smooth solutions for the compressible Navier–Stokes–Korteweg equations towards those for the incompressible Navier–Stokes equations with a convergence rate. As we know, it is the first result about zero Mach number limit of the compressible Navier–Stokes–Korteweg equations.

#### Keywords

compressible Navier–Stokes–Korteweg equations, Mach number limit, convergence-stability principle, incompressible Navier–Stokes equations, energy-type error estimates

#### 2010 Mathematics Subject Classification

35B40, 76W05

Published 16 September 2015