Convergence of a low order non-local Navier-Stokes-Korteweg system: the order-parameter model

TL;DR

This paper proves the global existence and convergence of solutions for a non-local Navier-Stokes-Korteweg system, simplifying numerical analysis by reducing derivative order and connecting it to classical models.

Contribution

It introduces and analyzes the order-parameter model, demonstrating its convergence to the local Korteweg system and establishing global well-posedness for near-equilibrium data.

Findings

Unique global solutions for initial data close to equilibrium.

Convergence of the non-local model to the local Korteweg system.

Reduction of numerical difficulties in capillary compressible systems.

Abstract

In the present article we consider a capillary compressible system introduced by C. Rohde after works of Bandon, Lin and Rogers, called the order-parameter model, and whose aim is to reduce the numerical difficulties that one encounters in the case of the classical local Korteweg system (involving derivatives of order three) or the non-local system (also introduced by Rohde after works of Van der Waals, and which involves a convolution operator). We prove that this system has a unique global solution for initial data close to an equilibrium and we precisely study the convergence of this solution towards the local Korteweg model.