Global refined Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes equations with large density

TL;DR

This paper establishes the global existence and control of solutions to the 3-D inhomogeneous incompressible Navier-Stokes equations with large density variations, improving previous smallness conditions and growth estimates.

Contribution

It proves the global existence of Fujita-Kato solutions under less restrictive initial data conditions and enhances the growth estimates to uniform-in-time bounds.

Findings

Global solutions exist under large density variations.

Initial data in critical Sobolev and Besov spaces suffice for control.

Improved exponential growth estimates for solutions.

Abstract

We investigate the global unique Fujita-Kato solution to the 3-D inhomogeneous incompressible Navier-Stokes equations with initial velocity $u_0$ being sufficiently small in critical spaces and with initial density being bounded from above and below. We first prove the global existence of Fujita-Kato solution to the system if we assume in addition that the initial velocity is in the critical Sobolev space. While under the additional assumptions that the initial velocity is in the critical Besov space and initial density is in a critical Besov space, we prove that the solutions are controlled by the norm of the initial data. Our results not only improve the smallness condition in the previous references for the initial velocity concerning the global Fujita-Kato solution of the system but also improve the exponential-in-time growth estimate for the solution in the paper [Abidi-Gui-Zhang, ARMA 2012] to be the uniform-in-time estimate.