Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable

TL;DR

This paper establishes the global well-posedness of 3-D inhomogeneous Navier-Stokes equations with large data that vary slowly in one direction, using critical Besov space techniques and nonlinear smallness conditions.

Contribution

It introduces a novel approach combining anisotropic Besov space analysis with smallness conditions to handle large, slowly varying initial data for the inhomogeneous Navier-Stokes system.

Findings

Proved global well-posedness under nonlinear smallness conditions.

Extended results to large data slowly varying in one direction.

Demonstrated consistency with anisotropic regularity propagation.

Abstract

In this paper, we are concerned with the global wellposedness of 3-D inhomogeneous incompressible Navier-Stokes equations \eqref{1.3} in the critical Besov spaces with the norm of which are invariant by the scaling of the equations and under a nonlinear smallness condition on the isentropic critical Besov norm to the fluctuation of the initial density and the critical anisotropic Besov norm of the horizontal components of the initial velocity which have to be exponentially small compared with the critical anisotropic Besov norm to the third component of the initial velocity. The novelty of this results is that the isentropic space structure to the homogeneity of the initial density function is consistent with the propagation of anisotropic regularity for the velocity field. In the second part, we apply the same idea to prove the global wellposedness of \eqref{1.3} with some large data which are slowly varying in one direction.