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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]

In this paper, we consider the 3D density-dependent magnetohydrodynamic equations with vacuum in the whole space R 3 , and provide a regularity criterion involving the velocity field in BMO space norm. This work generalizes the regularity criterion of the constant density MHD equations to the density-dependent one. [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]

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]

Tropical climate model derived by Frierson et al. (Commun Math Sci 2:591–626, 2004) and its modified versions have been investigated in a number of papers [see, e.g., Li and Titi (Discrete Contin Dyn Syst Series A 36(8):4495–4516, 2016), Wan (J Math Phys 57(2):021507, 2016), Ye (J Math Anal Appl 446:307–321, 2017) and more recently Dong et al. (Discrete Contin Dyn Syst Ser B 24(1):211–229, 2019)]. Here, we deal with the 2D tropical climate model with fractional dissipative terms in the equation of the barotropic mode u and in the equation of the first baroclinic mode v of the velocity, but without diffusion in the temperature equation, and we establish a regularity criterion for this system. [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 paper, we establish a new regularity criterion for weak solutions to the 3D incompressible MHD equations in terms of two pairs of (∂iui, ∂ibi)(i=1,2,3). More precisely, it is proved that the weak solution (u, b) is smooth on (0, T], provided that for some i, j ∈ {1, 2, 3} with i ≠ j, it holds that ∂iui,∂juj,∂ibi,∂jbj∈Lp0,T;B˙q,∞0(ℝ3),2p+3q=2,3≤q≤∞. [ABSTRACT FROM AUTHOR]

In this paper we establish a regularity criterion for the 2D full compressible MHD equations with zero heat conductivity and initial vacuum in a bounded domain. [ABSTRACT FROM AUTHOR]

The border basis scheme B O parametrizes all 0-dimensional ideals in K [ x 1 , ... , x n ] , where K is an arbitrary field, which have a border basis with respect to a given order ideal of terms O . Its vanishing ideal I (B O) is generated by quadratic equations which are easy to describe, and it contains a unique monomial point, namely the point corresponding to the monomial ideal generated by the terms outside O . Based on a detailed study of the generators of I (B O) , we describe a K-basis of the cotangent space m / m 2 , where m is the maximal ideal of the monomial point. Moreover, we provide an efficient algorithm to compute such a basis and use it to characterize the regularity of the monomial point. [ABSTRACT FROM AUTHOR]

In this work, we prove some regularity criteria for a Ginzburg‐Landau‐Navier‐Stokes in superfluidity in Rn(n≥3). [ABSTRACT FROM AUTHOR]

The paper deals with the regularity criteria for the weak solutions to the 3D Boussinesq equations in terms of the partial derivatives in Besov spaces. It is proved that the weak solution (u , θ) becomes regular provided that (∇ h u ~ , ∇ h θ) ∈ L 1 (0 , T ; B · ∞ , ∞ 0 (R 3)) Our results improve and extend the well-known result of Dong and Zhang (Nonlinear Anal 11:2415–2421, 2010) for the Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problems for certain regularized models to the incompressible viscoelastic flow in n space dimensions with n=2,3. Firstly, we establish a regularity condition for the solution under ∇u∈L1(0,T;L∞(Rn)). Furthermore, we obtain a regularity condition to the smooth solution for the inviscid regularized models in two space dimensions. Finally, we prove a global existence result of classical solutions for a three‐dimensional incompressible Oldroyd‐α model with fractional diffusion. [ABSTRACT FROM AUTHOR]

This work establishes a new regularity criterion for the 3D incompressible MHD equations in term of one directional derivative of the pressure (i.e., ∂ 3 P ) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that for T > 0 , if ∂ 3 P ∈ L β (0 , T ; L α (R x 1 x 2 2 ; L γ (R x 3))) with 2 β + 1 γ + 2 α = k ∈ [ 2 , 3) and 3 k ≤ γ ≤ α ≤ 1 k - 2 , then the corresponding solution (u, b) to the 3D MHD equations is regular on [0, T]. [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

In this work, we prove some regularity criteria for a Ginzburg–Landau–Navier–Stokes system with the Coulomb gauge in a bounded domain Ω ⊂ R 3 . [ABSTRACT FROM AUTHOR]

In this paper, we consider the regularity criteria, where a condition is added only on the velocity field to guarantee the smoothness of whole unknowns, and the global existence of the solutions with small initial data for the 3D tropical climate model with diffusion. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem to the 2D generalized tropical climate model. We establish a new blow-up criterion only in terms of the first baroclinic mode of the vector velocity in Besov spaces with negative indices. [ABSTRACT FROM AUTHOR]

This work establishes a new logarithmical improved regularity criterion for the 3D Boussinesq equations in terms of one directional derivative of the pressure (i.e., ∂ 3 P ) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that if ∫ 0 T ‖ ∂ 3 P (· , t) ‖ L x 3 γ L x 1 x 2 α β 1 + ln (1 + ‖ θ (· , t) ‖ L 4) d t < ∞ , where 2 β + 1 γ + 2 α = k ∈ [ 2 , 3) and 3 k ≤ γ ≤ α < 1 k - 2 , for some T > 0 , then the corresponding solution (u , θ) to the 3D Boussinesq equations is regular on [0, T]. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem of the Navier–Stokes–Maxwell system in three dimensions and establish two regularity criteria involving u (or ∇ u ) in Besov spaces and ∇ B in more flexible Lebesgue spaces. This improves and extends recent results of Ma, Jiang and Zhu. [ABSTRACT FROM AUTHOR]

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]

This paper deals with the asymptotic behavior, regularity criterion and global existence for the generalized Navier–Stokes equations. Firstly, an upper bound for the difference between the solution of our equation and the generalized heat equation in L 2 space is proved. We optimize the upper bound of decay for the solutions and obtain the algebraic lower bound by using Fourier splitting method. Then, a new scaling invariant regularity criterion on the fractional derivative is established. Finally, global existence is obtained provided that the initial data are small enough. [ABSTRACT FROM AUTHOR]

Two main results will be presented in our paper. First, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a $log$ supercritical assumption on the horizontally radial component $u^r$ and vertical component $u^z$, accompanied by a $log$ subcritical assumption on the horizontally angular component $u^θ$ of the velocity. Second, the precise Green function for the operator $-(Δ-\frac{1}{r^2})$ under the axially symmetric situation, where $r$ is the distance to the symmetric axis, and some weighted $L^p$ estimates of it will be given. This will serve as a tool for the study of axially symmetric Navier-Stokes equations. As an application, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a critical (or a subcritical) assumption on the angular component $w^θ$ of the vorticity. [ABSTRACT FROM AUTHOR]

We prove a regularity criterion for the 3D Navier-Stokes-Allen-Cahn system in a bounded smooth domain which improves the result obtained by Y. Li, S. Ding, and M. Huang [Discrete Contin. Dyn. Syst. Ser. B, 2016, 21(5): 1507–1523]. We also present a similar result to the 3D Navier-Stokes-Cahn-Hilliard system. [ABSTRACT FROM AUTHOR]

By means of the Littlewood‐Paley decomposition and the div‐curl Theorem by Coifman‐Lions‐Meyer‐Semmes, we prove an Osgood type regularity criterion for the 2D incompressible Oldroyd‐B model, that is, sup2≤N<+∞∫0T∑|j|≤N‖Δ˙jτ‖L∞NlogNds,where Δ˙j denotes the Fourier localization operator whose spectrum is supported in the shell {|ξ|≈2j}. [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]

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]

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]

In this paper, we establish several improved regularity criteria via partial components of vorticity to the incompressible Navier‐Stokes equations in higher dimensions. Our result is a further improvement of the recent result given by Tsurumi [extension criterion via partial components of vorticity on strong solutions to the Navier‐Stokes equations in higher dimensions, J. Differential Equations 263 (2017), 4007‐4022]. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Euler-Boussinesq equations in dimension two. Some new regularity criteria via the temperature are established by making use of the Littlewood-Paley technique and the new logarithmic Gronwall inequality. [ABSTRACT FROM AUTHOR]

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