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We prove that a smooth solution of a 3D model for electro-kinetic fluids in a bounded domain breaks down blows up at the same time as certain norm of vorticity. This norm is weaker than bmo-norm.

We obtain some regularity criteria for the wave map, a liquid crystals model, and the Hall-MHD with ion-slip effect.

In this article, we establish a regularity criterion for the Navier-Stokes system with axisymmetric initial data. It is proved that if the local axisymmetric smooth solution $u$ satisfies ${\|u^\theta\|_{L^{\alpha}(0,T; L^{\beta})}}

In this article, we consider the three-dimensional (3D) incompressible Boussinesq equations with zero thermal diffusion. We establish a regularity criterion for the local smooth solution in the framework of Besov spaces in terms of the velocity only.

In this article, we show an Osgood type regularity criterion for the three-dimensional Newton-Boussinesq equations, which improves the recent results in [4].

In this article, we consider the regularity for weak solutions to the Navier-Stokes equations in $mathbb{R}^3$. It is proved that if the horizontal derivatives of the two velocity components $$ abla _hwidetilde{u}in L^{2/(2-r)}(0,T;dot{mathcal{M}}_{2,3/r} (mathbb{R}^3)),quad hbox{for }0

In this note, we study the regularity of Leray-Hopf weak solutions to the Navier-Stokes equation, with the condition $$ abla (u_{1},u_{2},0) in L^{frac{2}{1-r}}(0,T; dot{mathcal{M}}_{2,3/r} (mathbb{R}^3) , $$ where $dot{mathcal{M}}_{2,3/r}(mathbb{R}^3)$ is the Morrey-Campanato space for $0

We consider the regularity of weak solutions to the Navier-Stokes equations in $mathbb{R}^3$. Let $u:=(u_1,u_2,u_3)$ be a weak solution and $widetilde{u}:=(u_1,u_2,0)$. We prove that $u$ is strong solution if $ ablawidetilde{u}$ satisfy Serrin's type criterion.

In this note, we study the regularity of Leray-Hopf weak solutions to the Navier-Stokes equation, with the condition ∇(u1, u2, 0) ∈ L2/1-r (0, T; Ṁ2,3/r(ℝ³), where Ṁ2,3/r(ℝ³) is the Morrey-Campanato space for 0 < r < 1. Since L1/3(ℝ³) ⊂ Ẋr(ℝ³) ⊂ &#x1E402,3/r(ℝ³), the above regularity condition allows us to improve the results obtained by Fan and Gao. [ABSTRACT FROM AUTHOR]