Inhomogeneous incompressible viscous flows with slowly varying initial data

TL;DR

This paper constructs a large class of initial data for the 3-D inhomogeneous Navier-Stokes equations, based on slowly varying functions, ensuring global smooth solutions by leveraging 2-D well-posedness and decay properties.

Contribution

It introduces a novel class of initial data based on slow variation in one direction, enabling global solutions for 3-D inhomogeneous Navier-Stokes equations.

Findings

Established global smooth solutions for a broad class of initial data.

Demonstrated optimal decay estimates for solutions similar to 2-D homogeneous cases.

Linked 3-D solutions to 2-D well-posedness and decay properties.

Abstract

The purpose of this paper is to provide a large class of initial data which generates global smooth solution of the 3-D inhomogeneous incompressible Navier-Stokes system in the whole space~$\R^3$. This class of data is based on functions which vary slowly in one direction. The idea is that 2-D inhomogeneous Navier-Stokes system with large data is globally well-posedness and we construct the 3-D approximate solutions by the 2-D solutions with a parameter. One of the key point of this study is the investigation of the time decay properties of the solutions to the 2-D inhomogeneous Navier-Stokes system. We obtained the same optimal decay estimates as the solutions of 2-D homogeneous Navier-Stokes system.