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# Communications in Mathematical Sciences

## Volume 21 (2023)

### Number 7

### High friction limits of Euler–Navier–Stokes–Korteweg equations for multicomponent models

Pages: 1815 – 1863

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n7.a4

#### Authors

#### Abstract

In this paper we analyze the high friction regime for the Navier–Stokes–Korteweg equations for multicomponent systems. According to the shape of the mixing and friction terms, we shall perform two limits: the high friction limit toward an equilibrium system for the limit densities and the *barycentric* velocity, and, after an appropriate time scaling, the diffusive relaxation toward parabolic, gradient flow equations for the limit densities. The rigorous justification of these limits is done by means of relative entropy techniques in the framework of weak, finite energy solutions of the relaxation models, rewritten in the enlarged formulation in terms of the *drift velocity*, toward smooth solutions of the corresponding equilibrium dynamics. Finally, since our estimates are uniform for small viscosity, the results are also valid for the Euler–Korteweg multicomponent models, and the corresponding estimates can be obtained by sending the viscosity to zero.

#### Keywords

high friction limit, diffusive relaxation, Euler–Navier–Stokes–Korteweg equations, Stefan–Maxwell systems, relative entropy method

#### 2010 Mathematics Subject Classification

35B25, 35L65

Received 30 March 2022

Received revised 6 December 2022

Accepted 13 January 2023

Published 9 October 2023