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

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 $\sup (\frac{1}{p},\frac{1}{q})\le \frac{1}{3}+\inf (\frac{1}{p},\frac{1}{q})$, we prove that if $C\Vert a_0{\Vert }_{{\dot{B}}_{q,1}^{\frac{3}{q}}}^{\alpha }(\Vert u_0^3{\Vert }_{{\dot{B}}_{p,1}^{-1+\frac{3}{p}}}/\mu +1)\le 1$, $\frac{C}{\mu }(\Vert u_0^h{\Vert }_{{\dot{B}}_{p,1}^{-1+\frac{3}{p}}}+\Vert u_0^3{\Vert }_{{\dot{B}}_{p,1}^{-1+\frac{3}{p}}}^{1-\alpha }\Vert u_0^h{\Vert }_{{\dot{B}}_{p,1}^{-1+\frac{3}{p}}}^{\alpha })\le 1$, then the system has a unique global solution $a\in \tilde{\mathcal{C}}([0,\infty );{\dot{B}}_{q,1}^{\frac{3}{q}}({\mathbb{R}}^3))$, $u\in \tilde{\mathcal{C}}([0,\infty );{\dot{B}}_{p,1}^{-1+\frac{3}{p}}({\mathbb{R}}^3))\cap L^1({\mathbb{R}}^{+};{\dot{B}}_{p,1}^{1+\frac{3}{p}}({\mathbb{R}}^3))$. 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.

18 pages

References in corpus (1)

• Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations