Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density

TL;DR

This paper establishes the global existence and uniqueness of solutions to the inhomogeneous Navier-Stokes equations with bounded initial density, improving previous results by relaxing regularity and smallness conditions on initial data.

Contribution

It proves global well-posedness for inhomogeneous Navier-Stokes equations with less restrictive initial velocity regularity and density fluctuation conditions, extending prior work.

Findings

Global existence and uniqueness of solutions for 2D and 3D cases.

Relaxed initial velocity regularity requirements compared to previous results.

Improved conditions on initial density fluctuation for global well-posedness.

Abstract

In this paper, we prove the global existence and uniqueness of solution to d-dimensional (for $d=2,3$) incompressible inhomogeneous Navier-Stokes equations with initial density being bounded from above and below by some positive constants, and with initial velocity $u_0\in H^s(\R^2)$ for $s>0$ in 2-D, or $u_0\in H^1(\R^3)$ satisfying $|u_0|_{L^2}|\na u_0|_{L^2}$ being sufficiently small in 3-D. This in particular improves the most recent well-posedness result in [10], which requires the initial velocity $u_0\in H^2(\R^d)$ for the local well-posedness result, and a smallness condition on the fluctuation of the initial density for the global well-posedness result.