A Lagrangian approach for the incompressible Navier-Stokes equations with variable density

TL;DR

This paper establishes the existence and uniqueness of global solutions for the inhomogeneous Navier-Stokes equations with variable density using a Lagrangian approach, under small initial data assumptions in a critical functional framework.

Contribution

It introduces a Lagrangian method to prove global well-posedness for variable density Navier-Stokes equations with small initial data, including piecewise constant densities.

Findings

Global existence and uniqueness of solutions under small data conditions

Piecewise constant densities with small interface jumps are admissible

Lagrangian coordinates facilitate the contraction mapping approach

Abstract

Here we investigate the Cauchy problem for the inhomogeneous Navier-Stokes equations in the whole $n$-dimensional space. Under some smallness assumption on the data, we show the existence of global-in-time unique solutions in a critical functional framework. The initial density is required to belong to the multiplier space of $\dot B^{n/p-1}_{p,1}(\R^n)$. In particular, piecewise constant initial densities are admissible data \emph{provided the jump at the interface is small enough}, and generate global unique solutions with piecewise constant densities. Using Lagrangian coordinates is the key to our results as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence.