Space-Time Regularity for the Three Dimensional Navier–Stokes and MHD Equations

Abstract

In this paper, we investigate the space-time regularity of solutions to (1) the three dimensional incompressible Navier–Stokes equations for initial data $u_{0}=(u_{0}^{h},u_{0}^{3}) \in \dot{B}_{p,r}^{ \frac{3}{p}-1} (\mathbb{R}^{3})$ with large initial vertical velocity component; and (2) the three dimensional incompressible magneto-hydrodynamic equations for initial datum $u_{0}=(u_{0}^{h},u _{0}^{3})\in \dot{B}_{p,r}^{\frac{3}{p}-1} (\mathbb{R}^{3})$ with large initial vertical velocity component and $b_{0}=(b_{0}^{h},b_{0}^{3}) \in \dot{B}_{p,r}^{\frac{3}{p}-1} (\mathbb{R}^{3})$ with large initial vertical magnetic field component.