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We establish the regularity criterion of weak solutions to the incompressible non‐Newtonian fluids with shear thinning viscosity in the whole space R3 for 85

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 establish a new regularity criterion in terms of the magnetic field for the local-in-time smooth solution of the two-dimensional non-resistive MHD equations. In particular, our result settles an open problem remarked by Lei et al. (2010), [11]. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a weak solution of the Cauchy problem for 3D unsteady flows of a generalized Newtonian fluid becomes a strong solution for 5 3 < p < 11 5 provided that the velocity field belongs to the critical space L 2 p − 1 (0 , T ; B M O (R 3)). The main key is to apply variants of Gagliardo–Nirenberg's inequality including a BMO -norm. In particular, when 2 < p < 11 5 , we derive and use a nonlinear variant of Gagliardo–Nirenberg's inequality including ‖ u ‖ B M O and ‖ | D u | p − 2 2 ∇ 2 u ‖ 2. [ABSTRACT FROM AUTHOR]

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 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 kkL;1(0;T;L1(R3)) with 2 + 3 = 2 and 32< < 1 or krkL;1(0;T;L;1(R3)) with 2 + 3 = 3 and 1 < 1 is suffciently small. [ABSTRACT FROM AUTHOR]

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 ... then the weak solution (u,w) is regular. [ABSTRACT FROM AUTHOR]

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 $.

We investigate regularity criteria for weak solutions to the Cauchy problem of the 3d co-rotational Beris-Edwards system for nematic liquid crystals, which couples the Navier–Stokes equations for the fluid velocity u with an evolution-diffusion equations for the Q -tenser. Our results yield that for any positive constant γ > 0 , if either the negative part of the associated pressure Π satisfies Π − [ ln ⁡ (1 + Π −) ] 1 + γ ∈ L ∞ (R + ; L 3 2 , ∞ (R 3)) , or the quantity 2 Π + | u | 2 + | ∇ Q | 2 satisfies (2 Π + + | u | 2 + | ∇ Q | 2) [ ln ⁡ (1 + 2 Π + + | u | 2 + | ∇ Q | 2) ] 1 + γ ∈ L ∞ (R + ; L 3 2 , ∞ (R 3)) , then the weak solution (u , Q) , to the 3d co-rotational Beris-Edwards system, is global-in-time smooth. Here, the subscript "−" and "+" denote the negative and the nonnegative part, respectively. L 3 2 , ∞ (R 3) denotes the standard weak Lebesgue space. If Q ≡ 0 , then our results extend some previous known results from the theory of the 3d Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper gives a further investigation on the regularity criteria for three-dimensional micropolar equations in Besov spaces. More precisely, it is proved that the weak solution (μ, ω) is regular if the velocity u satisfies. [ABSTRACT FROM AUTHOR]

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

Abstract This note obtains a new regularity criterion for the three-dimensional magneto-micropolar fluid flows in terms of one velocity component and the gradient field of the magnetic field. The authors prove that the weak solution ( u , ω , b ) $(u,\omega,b)$ to the magneto-micropolar fluid flows can be extended beyond time t = T $t=T$ , provided if u 3 ∈ L β ( 0 , T ; L α ( R 3 ) ) $u_{3}\in L^{\beta }(0,T;L^{\alpha }(R^{3}))$ with 2 β + 3 α ≤ 3 4 + 1 2 α , α > 10 3 $\frac{2}{\beta }+\frac{3}{\alpha }\leq \frac{3}{4}+\frac{1}{2\alpha },\alpha > \frac{10}{3}$ and ∇ b ∈ L 4 p 3 ( p − 2 ) ( 0 , T ; M ˙ p , q ( R 3 ) ) $\nabla b\in L^{\frac{4p}{3(p-2)}}(0,T;\dot{M}_{p,q}(R^{3}))$ with 1 < q ≤ p < ∞ $1< q\leq p

In this paper, we prove some new L^p-estimates of the velocity by the technique of L^p-energy method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we establish the regularity criterion for the weak solution of nematic liquid crystal flows in three dimensions when the L ∞ ( 0 , T ; B ˙ ∞ , ∞ − 1 ) $L^{\infty }(0,T;\dot{B}_{\infty,\infty }^{-1})$ -norm of a suitable low frequency part of ( u , ∇ d ) $(u,\nabla d)$ is bounded by a scaling invariant constant and the initial data ( u 0 , ∇ d 0 ) $(u_{0},\nabla d_{0})$ . Our result refines the corresponding one in (Liu and Zhao in J. Math. Anal. Appl. 407:557-566, 2013) and that in (Ri in Nonlinear Anal. TMA 190:111619, 2020).

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

This note obtains a new regularity criterion for the three-dimensional magneto-micropolar fluid flows in terms of one velocity component and the gradient field of the magnetic field. The authors prove that the weak solution (u , ω , b) to the magneto-micropolar fluid flows can be extended beyond time t = T , provided if u 3 ∈ L β (0 , T ; L α (R 3)) with 2 β + 3 α ≤ 3 4 + 1 2 α , α > 10 3 and ∇ b ∈ L 4 p 3 (p − 2) (0 , T ; M ˙ p , q (R 3)) with 1 < q ≤ p < ∞ and p ≥ 3 . [ABSTRACT FROM AUTHOR]

We prove a regularity criterion for a nonhomogeneous incompressible Ginzburg-Landau-Navier-Stokes system with the Coulomb gauge in ℝ3. It is proved that if the velocity field in the Besov space satisfies some integral property, then the solution keeps its smoothness. [ABSTRACT FROM AUTHOR]

Abstract This paper concentrates on the global regularity of classical solution to the 212 $2\frac{1}{2}$D magnetic Bénard system with partial dissipation, magnetic diffusion, and thermal diffusivity (i.e., horizontal dissipation, horizontal magnetic diffusion, and horizontal thermal diffusivity; vertical dissipation, vertical magnetic diffusion, and vertical thermal diffusivity). For the 212 $2\frac{1}{2}$D magnetic Bénard system with full dissipation, magnetic diffusion, and thermal diffusivity, the global existence and uniqueness can be obtained by the standard energy method. However, can the classical solution for the 212 $2\frac{1}{2}$D incompressible magnetic Bénard system still keep its global regularity when losing some partial dissipation, magnetic diffusion, and thermal diffusivity terms? We will give a rigorous proof to the global regularity for the 212 $2\frac{1}{2}$D magnetic Bénard system with horizontal and vertical dissipation, magnetic diffusion, and thermal diffusivity respectively in this paper. Furthermore, we also show that any possible finite time blow-up can be controlled by the L∞ $L^{\infty }$-norm of the vertical velocity and magnetic components, not include the temperature component (see Theorems 1.1 and 1.2). The results extend the recent work by Cheng and Du (J. Math. Fluid Mech. 17:769–797, 2015), and generalize the recent works by Regmi (Math. Methods Appl. Sci. 40:1497–1504, 2017), and Ma and Zhang (Bound. Value Probl. 2018:79, 2018).

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 paper, we establish the regularity criterion for the weak solution of nematic liquid crystal flows in three dimensions when the L ∞ (0 , T ; B ˙ ∞ , ∞ − 1) -norm of a suitable low frequency part of (u , ∇ d) is bounded by a scaling invariant constant and the initial data (u 0 , ∇ d 0) . Our result refines the corresponding one in (Liu and Zhao in J. Math. Anal. Appl. 407:557-566, 2013) and that in (Ri in Nonlinear Anal. TMA 190:111619, 2020). [ABSTRACT FROM AUTHOR]

Multiple kernel learning (MKL), as a principled classification method, selects and combines base kernels to increase the categorization accuracy of Support Vector Machines (SVMs). The group method of data handling neural network (GMDH-NN) has been applied in many fields of optimization, data mining, and pattern recognition. It can automatically seek interrelatedness in data, select an optimal structure for the model or network, and enhance the accuracy of existing algorithms. We can utilize the advantages of the GMDH-NN to build a multiple graph kernel learning (MGKL) method and enhance the categorization performance of graph kernel SVMs. In this paper, we first define a unitized symmetric regularity criterion (USRC) to improve the symmetric regularity criterion of GMDH-NN. Second, a novel structure for the initial model of the GMDH-NN is defined, which uses the posterior probability output of graph kernel SVMs. We then use a hybrid graph kernel in the H 1 -space for MGKL in combination with the GMDH-NN. This way, we can obtain a pool of optimal graph kernels with different kernel parameters. Our experiments on standard graph datasets show that this new MGKL method is highly effective. • We propose a unitized symmetric regularity criterion. • We define a novel structure of initial model of GMDH-NN. • We use graph kernel SVM as base learner and integrate it with GMDH. • We design a special MGKL approach based on the GMDH-NN. • We use several scoring sorting methodologies to evaluate our experiments. [ABSTRACT FROM AUTHOR]

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

Abstract We consider a Keller-Segel-Euler model and prove a regularity criterion of the local strong solutions in a 3D bounded domain Ω.

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

This paper proves a logarithmic regularity criterion for 3D Navier-Stokes system in a bounded domain with the Navier-type boundary condition. [ABSTRACT FROM AUTHOR]

Abstract This paper proves a regularity criterion for a new density-dependent incompressible Hall-MHD system with positive density. [ABSTRACT FROM AUTHOR]

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.

Abstract This paper is devoted to the Beale–Kato–Majda (BKM) regularity criterion of the 2D micropolar equations in the critical fractional diffusion situation α + β = 1 for α > 0 and β > 0. With the use of the Littlewood–Paley theory, the global existence of the unique solution (u , w) to the equations is found to be completely determined by the velocity field u. The dynamical behaviour of the scalar angular velocity w is controlled by the velocity u. More precisely, it is shown that the strong solution (u , w) exists globally if ∫ 0 T ‖ ∇ × u (τ) ‖ L ∞ 2 d τ < ∞ for any T < ∞. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we use the bootstrap argument to prove the uniform-in- k global strong solutions of the 2D density-dependent incompressible magnetic Bénard problem in a bounded domain. Here k is the heat conductivity coefficient. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for the 3D Leray-α-MHD model is investigated. We obtain the logarithmically improved blow-up criterion of smooth solutions for the Leray-α-MHD model in terms of the magnetic field B only in the framework of homogeneous Besov space with negative index. [ABSTRACT FROM AUTHOR]

Abstract In this paper we consider the 3D compressible MHD system in a bounded domain. We first prove a regularity criterion and then use it and the bootstrap argument to show the uniform-in- η global small solutions. Here η is the resistivity coefficient. [ABSTRACT FROM AUTHOR]

Abstract This work establishes a new regularity criterion for the 3D incompressible magneto-hydrodynamical (MHD) equations in terms of one directional derivative of the velocity (i.e., ∂ 3 u) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that if ∂ 3 u satisfies ∫ 0 T ‖ ‖ ∂ 3 u (τ) ‖ L x 3 α ‖ L x 1 x 2 β q d τ < + ∞ , where 2 q + 1 α + 2 β = k ∈ [ 1 , 3 2 ] and 3 k < α ≤ β ≤ 1 k − 1 , for some T > 0 , then the corresponding solution (u , b) to the 3D MHD equations is regular on (0 , T) × R 3. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we establish a new regularity criterion for the two-dimensional incompressible generalized magnetohydrodynamics equations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the two-dimensional incompressible generalized magnetohydrodynamics equations. We obtain some regularity criteria to the corresponding equations in terms of the current density j or the vorticity ω. Our approach is based on a key observation on the nice structure of the incompressible magnetohydrodynamics equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the axisymmetric Navier–Stokes equations with non-zero swirl component. It is proved that if r u r ≥ M for some M > − 2 , then the solution actually is smooth. This extends the result in Pan (2017) [14] . [ABSTRACT FROM AUTHOR]

The paper is dedicated to study of the regularity criterion for weak solutions to the 3D incompressible MHD equations. Employing the Littlewood–Paley decomposition, we show that if ∇ ˜ u ˜ = ( ∂ 1 u ˜ , ∂ 2 u ˜ ) ∈ L s 1 ( [ 0 , T ) ; B ˙ r 1 , 2 r 1 3 0 ( R 3 ) ) , 2 s 1 + 3 r 1 = 2 , 3 2 < r 1 ≤ ∞ and ∇ ˜ b ˜ = ( ∂ 1 b ˜ , ∂ 2 b ˜ ) ∈ L s 2 ( [ 0 , T ) ; B ˙ r 2 , 2 r 2 3 0 ( R 3 ) ) , 2 s 2 + 3 r 2 = 2 , 3 2 < r 2 ≤ ∞ , then the solutions to the MHD actually is smooth on ( 0 , T ) . [ABSTRACT FROM AUTHOR]