Global solutions to the Navier--Stokes equations with large vertical velocities in $\dot{B}_{\infty,σ}^{-1}(\mathbb{R}^3)$

Abstract

In this paper, we consider the Cauchy problem for the $3$D incompressible Navier--Stokes equations and prove the existence of unique global solutions in the framework that the horizontal component of the velocity field is small in some critical Besov spaces including the classical Fujita--Kato class, while the vertical component is large in the wide class $\dot{B}_{\infty,σ}^{-1}(\mathbb{R}^3)$ ($1 \leq σ< \infty$) where the Navier--Stokes equations are known to be ill-posed in.