Global Solutions to the 3-D Incompressible Inhomogeneous Navier–Stokes System with Rough Density

Abstract

In this paper, we prove the global well-posedness of the 3-D incompressible inhomogeneous Navier–Stokes equations with initial data $a_{0} \in \mathcal{M}(B_{p,1}^{-1+ \frac{3} {p} }({\mathbb{R}}^{3})),$ $u_{0} = (u_{0}^{h},u_{0}^{3}) \in B_{p,1}^{-1+ \frac{3} {p} }({\mathbb{R}}^{3}),$ which satisfies

$$(\mu \|a_{0}\|_{ \mathcal{M}(B_{p,1}^{-1+ \frac{3} {p} })} +\| u_{0}^{h}\|_{ B_{p,1}^{-1+ \frac{3} {p} }})\exp (C_{0}\|u_{0}^{3}\|_{ B_{p,1}^{-1+ \frac{3} {p} }}^{2}\ /{\mu }^{2}) \leq c_{ 0}\mu $$

for some positive constants c ₀, C ₀ and $\frac{3} {2} < p < 6.$ The novelty of this paper is to replace $\|a_{0}\|_{ B_{q,1}^{ \frac{3} {q} }}$ in the smallness condition of [20] by the rough norm in the multiplier space $\|a_{0}\|_{ \mathcal{M}(B_{p,1}^{-1+ \frac{3} {p} })}$ here.

2010 Mathematics Subject Classification: 35Q30, 76D03.