On a Lagrangian method for the convergence from a non-local to a local Korteweg capillary fluid model

TL;DR

This paper investigates the convergence of a non-local capillarity model to a local Korteweg system for compressible fluids, using a Lagrangian approach to derive precise energy estimates and analyze the capillarity effects.

Contribution

It introduces a Lagrangian method to analyze the convergence from non-local to local Korteweg models, providing detailed energy estimates in hybrid Besov spaces.

Findings

Proved convergence of solutions from non-local to local Korteweg system.

Developed energy estimates in hybrid Besov spaces based on frequency thresholds.

Analyzed the impact of Lagrangian change on non-local capillary terms.

Abstract

In the present article we are interested in further investigations for the barotropic compressible Navier-Stokes system endowed with a non-local capillarity we studied in [7]. Thanks to an accurate study of the associated linear system using a Lagrangian change of coordinates, we provide more precise energy estimates in terms of hybrid Besov spaces naturally depending on a threshold frequency (which is determined in function of the physical parameter) distinguishing the low and the high regimes. It allows us in particular to prove the convergence of the solutions from the non-local to the local Korteweg system. Another mathematical interest of this article is the study of the effect of the Lagrangian change on the non-local capillary term.