Between homogeneous and inhomogeneous Navier-Stokes systems: the issue of stability

TL;DR

This paper constructs large velocity solutions for the 3D inhomogeneous Navier-Stokes system, demonstrating stability of 2D solutions with constant density under small initial density variations, using maximal regularity and Lagrangian coordinates.

Contribution

It introduces a novel stability analysis for inhomogeneous Navier-Stokes solutions in three dimensions, extending understanding of solution behavior near constant density states.

Findings

Established stability of 2D solutions in 3D setting.

Constructed large velocity solutions under small density perturbations.

Utilized maximal regularity estimates and Lagrangian framework.

Abstract

We construct large velocity vector solutions to the three dimensional inhomogeneous Navier-Stokes system. The result is proved via the stability of two dimensional solutions with constant density, under the assumption that initial density is point-wisely close to a constant. Key elements of our approach are estimates in the maximal regularity regime and the Lagrangian coordinates. Considerations are done in the whole $\R^3$.