Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 89, pp. 1-9.

Title: Logarithmically improved regularity criteria for the Navier-Stokes equations 
       in homogeneous Besov spaces
       
Authors: Nguyen Anh Dao (Univ. of Economics Ho Chi Minh City, Viet Nam)
         Jesus Ildefonso Diaz (Univ. Complutense de Madrid, Spain)

Abstract:
We investigate a logarithmically improved regularity criteria in terms of the velocity,
or 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].

Submitted August 03, 2021. Published November 03, 2021.
Math Subject Classifications: 35Q35, 35B65, 76D05.
Key Words: Besov space; Navier-Stokes equations; regularity criteria.