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J. Q. Yang (2019) established a regularity criterion for the 3D shear thinning fluids in the whole space $\Bbb{R}^3$ via two velocity components. The goal of this short note is to extend this result in viewpoint of Lorentz space.

In this paper, we obtained the global existence and large time behavior of the solution for bipolar Navier-Stokes-Poisson equations under the partially smallness assumption of the initial data. Due to the complexity of bipolar Navier-Stokes-Poisson equations, we chose Green's function method instead of the classical energy method and thus discussed the regularity criterion under decaying structures in time instead of only integrability of time variable. It made the whole proof more simple and clear, meanwhile, resulted in the large time decaying estimates of the solution. It also showed the advantage of Green's function method in the study of global existence in the large perturbation framework.

In this paper, we investigate the regularity criterion of weak solutions to three-dimensional magneto-micropolar fluid equations with fractional dissipation. A regularity criterion is established via the third component of the velocity fields, the micro-rotational velocity fields, and the magnetic fields.

In this article, we study the regularity criteria of the weak solutions to the Boussinesq equations involving the horizontal component of velocity or the horizontal derivatives of the two components of velocity in anisotropic Lorentz spaces. This result reveals that the velocity field plays a dominant role in regularity theory of the Boussinesq equations.

This paper is devoted to investigating regularity criteria for the 3D micropolar fluid equations in terms of pressure in weak Lebesgue space. More precisely, we prove that the weak solution is regular on $ (0, T] $ provided that either the norm $ \left\Vert \pi \right\Vert _{L^{\alpha, \infty }(0, T;L^{\beta, \infty }(\mathbb{R}^{3}))} $ with $ \frac{2}{\alpha }+ \frac{3}{\beta } = 2 $ and $ \frac{3}{2} < \beta < \infty $ or $ \left\Vert \nabla \pi \right\Vert _{L^{\alpha, \infty }(0, T;L^{\beta, \infty }(\mathbb{R} ^{3}))} $ with $ \frac{2}{\alpha }+\frac{3}{\beta } = 3 $ and $ 1 < \beta < \infty $ is sufficiently small.

This work focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed pressure-velocity-magnetic field in view of Lorentz spaces. Our main result shows the weak solution is regular, provided that π e−|x| 2 + |u| + |b| θ ∈ L p (0, T; L q,∞), where 2 p + 3 q = 2 − θ and 0 ≤ θ ≤ 1.

In this paper, we obtain a regularity criterion via horizontal components of velocity and molecular orientations for the 3D nematic liquid crystal flows. That is the smooth solution $ (u, d) $ can be extended beyond T, provided that $ \int_{0}^{T}(||u_{h}||_{\dot{B}_{\infty, \infty}^{0}}^{2}+||\nabla d||_{\dot{B}_{\infty, \infty}^{0}}^{2}) \mbox{d} t < \infty $.

In this paper, we establish a regularity criterion for micropolar fluid flows in terms of the one component of the velocity in critical Morrey-Campanato space. More precisely, we show that if where then the weak solution (u,w) is regular.

In this paper, we establish a regularity criterion for the 3D nematic liquid crystal flows. More precisely, we prove that the local smooth solution $ (u, d) $ is regular provided that velocity component $ u_{3} $, vorticity component $ \omega_{3} $ and the horizontal derivative components of the orientation field $ \nabla_{h}d $ satisfy $ \begin{eqnarray*} \int_{0}^{T}|| u_{3}||_{L^{p}}^{\frac{2p}{p-3}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}}+||\nabla_{h} d||_{L^{a}}^{\frac{2a}{a-3}} \mbox{d} t

Consider the time-periodic viscous incompressible fluid flow past a body with non-zero velocity at infinity. This article gives sufficient conditions such that weak solutions to this problem are smooth. Since time-periodic solutions do not have finite kinetic energy in general, the well-known regularity results for weak solutions to the corresponding initial-value problem cannot be transferred directly. The established regularity criterion demands a certain integrability of the purely periodic part of the velocity field or its gradient, but it does not concern the time mean of these quantities.

The aim of this paper is to establish the regularity criterion of weak solutions to the 3D micropolar fluid equations by one directional derivative of the pressure in anisotropic Lebesgue spaces. We improve the regularity criterion for weak solutions previously given by Jia, Zhang and Dong in [21].

In this paper we investigate the regularity criterion for the local-in-time smooth solution to the three-dimensional (3D) tropical climate model in the Morrey--Campanato space. It is shown that if $u $ satisfies \[\int^{T}_{0}\frac{\|\nabla u(t)\|^{\frac{2}{r}}_{\dot{\mathcal{M}}_{2,3/r}}}{\ln(\|u(t)\|_{L^2}+e)}dt

In this work, we investigate the regularitycriterion for the solution of the Hall-MHD system in three-dimensions. It isproved that if the pressure $\pi $ and the gradient of magnetic field $%\nabla B$ satisfies some kind of space-time integrable condition on $[0,T]$,then the corresponding solution keeps smoothness up to time $T$. This resultimproves some previous works to the Morrey space $\overset{\cdot }{\mathcal{M}}_{2,\frac{3}{r}}$ for $0\leq r

In this note, a regularity criterion of weaksolutions to the 3D-Boussinesq equations with respect to Serrin type condition under the framework of Besov space $\overset{.}{B}_{\infty ,\infty}^{r}$. It is shown that the weak solution $(u,\theta )$ is regular on $%(0,T] $ if $u$ satisfies $\int\limits_{0}^{T}{\left\| u\left( \cdot ,t \right) \right\|_{\overset{\cdot R}{\mathop{{{B}_{\infty ,\infty }}}}\,}^{\frac{2}{1+r}}}\ \ dt < \infty ,$ for 0

In this paper, logarithmically improved regularity criteria for the Boussinesq equations are established under the framework of Besov space $\overset{.}{B}_{\infty ,\infty }^{-r}$. We prove the solution $(u,\theta )$ is smooth up to time $T>0$ provided that \begin{equation} \int_{0}^{T}\frac{\left\Vert u(\cdot ,t)\right\Vert _{\overset{.}{B} _{\infty ,\infty }^{-r}}^{\frac{2}{1-r}}}{\log (e+\left\Vert u(t,.)\right\Vert _{\overset{.}{B}_{\infty ,\infty }^{-r}})}dt

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.

This paper establishes a regularity criterion for the strong solution to the Leray‐α‐MHD in Campanato‐Morrey space. We show that if the magnetic field b satisfies the following condition: ∫0T‖∇b(τ,·)‖M˙qp2p2p−3+‖D2b(τ,·)‖M˙qp2p2p−3dτ<∞, the strong solution (v,b) to the 3‐D Leray‐α MHD equation can be extended beyond T > 0. We also prove that if the magnetic field b satisfies ∫0T‖∇b(τ,·)‖M˙qppp−1dτ<∞, the strong solution (v,b) to the 2‐D Leray‐α MHD equation can be extended beyond T > 0, which extends the results in previous studies. [ABSTRACT FROM AUTHOR]

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.

Abstract: We consider the regularity problem for 3D Navier–Stokes equations in a bounded domain with smooth boundary. A new sufficient condition which guarantees the regularity of weak solutions on the quotient for the Navier–Stokes equations is established. [Copyright &y& Elsevier]

Abstract: In this work, a regularity criterion is proved for local strong solutions of the Navier–Stokes equations in the presence of mass diffusion. [Copyright &y& Elsevier]

Abstract: In this note, by using a well-known commutator estimate, we give a new regularity criterion for the 2D dissipative quasi-geostrophic equations in the critical and supercritical cases. [Copyright &y& Elsevier]

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]

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]

In this paper we consider the three-dimensional Navier-Stokes equations in infinite channel. We provide a regularity criterion for solutions of the three-dimensional Navier-Stokes equations in terms of the vertical component of the velocity field. [ABSTRACT FROM AUTHOR]

Abstract: We consider the regularity of axisymmetric weak solutions to the Navier–Stokes equations in . Let u be an axisymmetric weak solution in , , and be the azimuthal component of w in the cylindrical coordinates. Chae–Lee [D. Chae, J. Lee, On the regularity of axisymmetric solutions of the Navier–Stokes equations, Math. Z. 239 (2002) 645–671] proved the regularity of weak solutions under the condition , with , . We deal with the marginal case which they excluded. It is proved that u becomes a regular solution if . [Copyright &y& Elsevier]