Logarithmically improved regularity criteria for the Navier-Stokes equations in homogeneous Besov spaces

Abstract

We investigate a logarithmically improved regularity criteria in terms of the velocity, for the vorticity, for the Navier-Stokes equations in homogeneous Besov spaces. More precisely, we prove that if the weak solution u satisfies either $$\displaylines{ \int^T_0 \frac{\|u(t)\|^{\frac{2}{1-\alpha}}_{{\rm \dot{B}^{-\alpha}_{\infty, \infty}}}} {1+\log^+\|u(t)\|_{\dot{H}^{s_0}}} \, dt <\infty, \quad \text{or}\quad \int^T_0 \frac{\|w(t)\|_{\dot{B}^{-\alpha}_{\infty, \infty}}^\frac{2}{2-\alpha} } {1 + \log^ + \|w(t)\|_{\dot{H}^{s_0}}}\,dt<\infty\,, }$$ where w =rot u, then u is regular on (0,T].Our conclusions improve some results by Fan et al. [5]. For more information see https://ejde.math.txstate.edu/Volumes/2021/89/abstr.html