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

TL;DR

This paper proves the global well-posedness of the 3D inhomogeneous incompressible Navier-Stokes equations for large initial velocities under certain Besov space conditions, extending previous smallness results to polynomial conditions.

Contribution

It introduces a polynomial smallness condition on initial data ensuring global solutions, improving upon prior exponential smallness criteria.

Findings

Established global existence and uniqueness under new polynomial smallness conditions.

Extended previous results to larger initial velocities with less restrictive assumptions.

Demonstrated solutions remain in specified Besov spaces for all time.

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.