Regularity criterion for 3D Navier-Stokes Equations in Besov spaces

TL;DR

This paper establishes new regularity criteria for 3D Navier-Stokes solutions based on Besov space conditions on velocity gradients, including anisotropic and boundary conditions, advancing understanding of solution regularity.

Contribution

It introduces novel regularity criteria involving Besov space norms for velocity gradients and anisotropic conditions, extending prior criteria for Navier-Stokes solutions.

Findings

Weak solutions become regular under Besov space gradient conditions

Anisotropic regularity criteria are established for solutions in D space

Additional regularity criteria involve conditions on 3u_3

Abstract

Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. The results show that a weak solution $u$ becomes regular if the gradient of velocity component $\nabla_{h}{u}$ (or $ \nabla{u_3}$) satisfies the additional conditions in the class of $L^{q}(0,T; \dot{B}_{p,r}^{s}(\mathbb{R}^{3}))$, where $\nabla_{h}=(\partial_{x_{1}},\partial_{x_{2}})$ is the horizontal gradient operator. Besides, we also consider the anisotropic regularity criterion for the weak solution of Navier-Stokes equations in $\mathbb{R}^3$. Finally, we also get a further regularity criterion, when give the sufficient condition on $\partial_3u_3$.