Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity (1410.6300v1)

Abstract: In this article, we consider the global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity. More precisely, assuming $a_0 \in \dot{B}{q,1}^{\frac{3}{q}}(\mathbb{R}^3)$ and $u_0=(u_0^h,u_0^3)\in \dot{B}{p,1}^{-1+\frac{3}{p}}(\mathbb{R}^3)$ for $p,q \in (1,6)$ with $\sup(\frac{1}{p}, \frac{1}{q})\leq\frac{1}{3}+ \inf (\frac{1}{p}, \frac{1}{q})$, we prove that if $C|a_0|{\dot{B}{q,1}^{\frac{3}{q}}}^{\alpha}(|u_0^3|{\dot{B}{p,1}^{-1+\frac{3}{p}}}/{\mu}+1)\leq1$, $\frac{C}{\mu}(|u_0^h|{\dot{B}{p,1}^{-1+\frac{3}{p}}}+|u_0^3|{\dot{B}{p,1}^{-1+\frac{3}{p}}}^{1-\alpha}|u_0^h|{\dot{B}{p,1}^{-1+\frac{3}{p}}}^{\alpha})\leq 1$, then the system has a unique global solution $a\in\widetilde{\mathcal{C}}([0,\infty);\dot{B}{q,1}^{\frac{3}{q}}(\mathbb{R}^3))$, $u\in\widetilde{\mathcal{C}}([0,\infty);\dot{B}{p,1}^{-1+\frac{3}{p}}(\mathbb{R}^3))\cap L^1(\mathbb{R}^+;\dot{B}_{p,1}^{1+\frac{3}{p}}(\mathbb{R}^3))$. It improves the recent result of M. Paicu, P. Zhang (J. Funct. Anal. 262 (2012) 3556-3584), where the exponent form of the initial smallness condition is replaced by a polynomial form.