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In this paper, we show that a weak solution to 3D shear-thinning flows becomes strong provided that the gradient of one component of velocity belongs to a certain Lebesgue space class. [ABSTRACT FROM AUTHOR]

The goal of this paper is to establish a regularity criterion for the three-dimensional incompressible Hall-magnetohydrodynamic equations involving only the vorticity. Our result complements and improves some existing results. [ABSTRACT FROM AUTHOR]

In this paper, we give a BKM-type blow-up criterion, which involves only a single component of the velocity, for the inviscid axially symmetric Boussinesq system. More precisely, we will show that if the vorticity of the swirl part of the velocity belongs to L 1 (0 , T ∗ , L ∞) , then the solution is regular up to time T ∗ . At present, our results can not be easily generalized to the classical L t 1 B M O regularity criterion, which will be considered in our further works. [ABSTRACT FROM AUTHOR]

This paper focuses on the n-dimensional ( n = 2 , 3 ) Camassa–Holm equations with non-local diffusion of type (- Δ) s . In Gan et al. (Calc Var Partial Differ Equ. 60, 2021), they proved that with regular initial data, the finite energy weak solutions are indeed regular for all time if n / 4 < s < 1 . The purpose of this paper is to improve the result of Gan et al. Actually, we will establish the following regularity criterion: If ∇ u ∈ L q ([ 0 , T ] ; L r (R n)) with n r + 2 s q ≤ 2 s , r > n 2 s , then v is regular. As a corollary, we can obtain that if n - 2 4 < s < 1 , then the finite energy weak solutions of the fractional Camassa–Holm equations are regular. In addition, by partially appealing to some ideas used in partial regularity theory, when n = 3 , we can prove that if 1 2 < s < 1 , then the suitable weak solution of the fractional Camassa–Holm equations must be bounded. [ABSTRACT FROM AUTHOR]

This paper devotes to establish an improved regularity criterion for weak solution of 3D incompressible MHD equations. We show that the weak solution is regular, provided that ∫ 0 T ‖ π (· , t) ‖ B ˙ ∞ , ∞ - 1 2 e + ln e + ‖ π (· , t) ‖ B ˙ ∞ , ∞ - 1 ln e + ln e + ‖ π (· , t) ‖ B ˙ ∞ , ∞ - 1 d t < ∞. As a consequence, this improves some previous works. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the regularity criterion for weak solutions to the 3D incompressible MHD equations. We show that if any two groups functions of $$(\partial _{1}u_{1}, \partial _{1}b_{1})$$ , $$(\partial _{2}u_{2}, \partial _{2}b_{2})$$ and $$(\partial _{3}u_{3}, \partial _{3}b_{3})$$ belong to the space $$L^{\theta }([0,T);L^{r}(\mathbb {R}^3)), \frac{2}{\theta }+\frac{3}{r}=2, \frac{3}{2}

The paper deals with the 3D incompressible MHD equations and aims at improving a regularity criterion in terms of the horizontal gradient of velocity and magnetic field. It is proved that the weak solution ( u, b) becomes regular provided that The result is an extension of regularity criterion for 3D Navier-Stokes equations in Besov space due to Fang and Qian (Commun Pure Appl Anal 13:585-603, 2014) [see also (Ni et al. in J Math Anal Appl 396:108-118, 2012)]. [ABSTRACT FROM AUTHOR]

This paper focuses on the global regularity problem for a family of active scalar equations with fractional dissipation. We obtain an explicit lower bound on the local existence of solutions, the eventual regularity of solutions and the global regularity of solutions for large initial data for the supercritical active scalar equations. In addition, a new regularity criterion is presented for the supercritical case which is used to study the eventual regularity. [ABSTRACT FROM AUTHOR]

In this paper, some improved regularity criteria for the 3D viscous MHD equations are established in multiplier spaces. It is proved that if the velocity field satisfies or the gradient field of velocity satisfies then the solution remains smooth on [0, T]. [ABSTRACT FROM AUTHOR]