Lagragian methods for a general inhomogeneous incompressible Navier-Stokes-Korteweg system with variable capillarity and viscosity coefficients

TL;DR

This paper develops Lagrangian methods to establish local well-posedness for a complex inhomogeneous Navier-Stokes-Korteweg system with variable coefficients, and explores the limit behavior as capillarity effects diminish.

Contribution

It introduces a novel Lagrangian approach to analyze the inhomogeneous Navier-Stokes-Korteweg system with variable coefficients, extending well-posedness results to broader settings.

Findings

Local well-posedness in critical Besov spaces for variable coefficients

Lifespan tends to infinity as capillarity coefficient approaches zero

Connection to global existence results for constant coefficient systems

Abstract

We study the inhomogeneous incompressible Navier-Stokes system endowed with a general capillary term. Thanks to recent methods based on Lagrangian change of variables, we obtain local well-posedness in critical Besov spaces (even if the integration index p is different from 2) and for variable viscosity and capillary terms. In the case of constant coefficients and for initial data that are perturbations of a constant state, we are able to prove that the lifespan goes to infinity as the capillary coefficient goes to zero, connecting our result to the global existence result obtained by Danchin and Mucha for the incompressible Navier-Stokes system with constant coefficients.