Global Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes system

TL;DR

This paper proves the global existence of weak solutions for the 3D inhomogeneous incompressible Navier-Stokes system under specific initial conditions, extending classical results to a more general inhomogeneous setting.

Contribution

It establishes the global existence of weak solutions in the critical Besov space for inhomogeneous Navier-Stokes equations, generalizing Fujita-Kato solutions.

Findings

Global weak solutions exist with initial density bounded away from zero.

Method applies to cases with large initial velocity components.

Provides lower bounds for lifespan of smooth solutions.

Abstract

In this paper, we shall prove the global existence of weak solutions to 3D inhomogeneous incompressible Navier-Stokes system $({\rm INS})$ with initial density in the bounded function space and having a positive lower bound and with initial velocity being sufficiently small in the critical Besov space, $\dot B^{\f12}_{2,1}.$ This result corresponds to the Fujita-Kato solutions of the classical Navier-Stokes system. The same idea can be used to prove the global existence of weak solutions in the \emph{critical functional framework} to $({\rm INS})$ with one component of the initial velocity being large and can also be applied to provide a lower bound for the lifespan of smooth enough solutions of $({\rm INS}).$