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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 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]

We consider the three-dimensional inviscid Boussinesq–Voigt system which is a regularization model for the inviscid Boussinesq equations. We prove a regularity criterion for the weak solutions (in particular, for the time-derivative of the velocity), in L p -spaces, involving first derivatives of the pressure. [ABSTRACT FROM AUTHOR]

In this remark, we consider regularity criterion for weak solutions to the 3d incompressible Navier–Stokes equations via pressure. It is proved that if the corressponding pressure P satisfies ∇ h ˜ P ∈ L β (0 , T ; L p (R x 1 ; L q (R x 2 ; L r (R x 3)))) with 2 β + 1 p + 1 q + 1 r = 3 , 6 5 < p , q , r ≤ 3 and 2 − (1 p + 1 q + 1 r) ≥ 0 , then the weak solution remains smooth on (0 , T ] . Here, ∇ h ˜ = (0 , ∂ 2 , ∂ 3) . [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]

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 $.

By developing the local energy bound with B ˙ ∞ , ∞ - 1 and fully using the property of B ˙ ∞ , c (N) - 1 and B ˙ p , ∞ - 1 + 3 / p ∩ B ˙ ∞ , q - 1 (p , q < ∞) , we provide a blow-up criterion to the 3D Navier–Stokes equations in the critical Besov space with - 1 negative regularity. More precisely, if u 3 ∈ L ∞ (0 , T ; B ˙ p , ∞ - 1 + 3 / p ∩ B ˙ ∞ , q - 1) and u h ∈ L ∞ (0 , T ; B ˙ ∞ , ∞ - 1) with u h (T) ∈ B ˙ ∞ , c (N) - 1 , the smooth solution u = (u h , u 3) can be extended beyond T, which improves the result of Wang-Zhang (Sci China Math 60: 637–650, 2017) and Li-Zhou (Dyn PDE 18: 71–80, 2021). [ABSTRACT FROM AUTHOR]

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]

In this paper, we obtain the nth-logarithmically improved regularity criterion of smooth solutions for the incompressible Leray-\alpha-MHD model in terms of the magnetic field B. Meanwhile, the new logarithmically improved regularity criterion for the 3D Navier-Stokes equation in terms of the pressure \pi and gradient of velocity \nabla u can also be established. Especially, we explore a new logarithmically improved Serrin's criterion for the 3D Navier-Stokes equations, which improves the results of Lei and Zhou [Commn. Pure Appl. Anal. 12 (2013), pp. 2715–2719]. [ABSTRACT FROM AUTHOR]

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

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 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]

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]

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]

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 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]

Abstract: We prove a regularity criterion for an Ericksen‐Leslie's model when the space dimension n = 3. We also prove the global well‐posedness of strong solutions for a generalized wave map in a bounded domain when n = 1, and prove a regularity criterion for the n D generalized wave maps when n ≥ 2. [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]

In this paper, we consider the two-dimensional dissipative surface quasi-geostrophic equation, and establish a logarithmically improved regularity criterion. Consequently, our result extends the regularity criterion result of Dong and Pavlovié (2009). In addition, a logarithmically improved regularity criterion to the three-dimensional Navier–Stokes equations is also derived by the same arguments. Therefore, this result extends and improves many previous works. [ABSTRACT FROM AUTHOR]

In this paper we consider the three dimensional (3D) incompressible Hall-magnetohydrodynamic equations (Hall-MHD). It is proved that the local smooth solution can be continued beyond timeTprovided that the velocityuand the magnetic fieldBsatisfy for. As a consequence, this result extend the previous result. [ABSTRACT FROM PUBLISHER]

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 proves some regularity criteria for the 3D (density-dependent) incompressible Maxwell–Navier–Stokes system, which improves a recent result of Kang and Lee (2013). [ABSTRACT FROM AUTHOR]

This paper proves a regularity criterion for the $3D$ full compressible Navier-Stokes-Maxwell system in a bounded domain. [ABSTRACT FROM AUTHOR]

We prove two new regularity criteria for the 3D incompressible Navier-Stokes equations in a bounded domain. Our results also hold for the 3D Boussinesq system with zero heat conductivity. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new regularity criterion for the 3D Navier–Stokes equations in terms of only two vorticity components. By making use of a logarithmic Sobolev inequality in the Besov spaces with negative indices and well known commutator estimate, we prove that a unique local strong solution does not blow-up at time T if two components of vorticity belong to L 2 (0 , T ; B ̇ ∞ , ∞ − 1). This result is the further extension of the previous works by Guo et al. (2018) and by O (2021), and becomes the improvement of the work by Gala (2011). • A new regularity criterion via two vorticity components in the scaling invariant space. • Giving the satisfactory answer to the open question posed in previous paper. • Very simple and self-contained proof of the regularity. [ABSTRACT FROM AUTHOR]

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.

This paper is concerned with the short time strong solutions for Cauchy problem to a simplified Ericksen-Leslie system of compressible nematic liquid crystals in two dimensions with vacuum as far field density. We establish a blow-up criterion for possible breakdown of such solutions at a finite time, which is analogous to the well-known Serrin's blow-up criterion for the incompressible Navier-Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper establishes a new regularity criterion for the Navier-Stokes equations in terms of two velocity components. We show that if the two velocity components ũ = (u1, u2, 0) satisfy ∫T0 ∥ũ(s)∥2B0∞,∞ds < ∞, then the solution can be smoothly extended after t = T. This gives an answer to an open problem in [B. ǫ. Dong, Z. Zhang, Nonlinear Anal. Real World Appl. 11(2010), 2415-2421]. [ABSTRACT FROM AUTHOR]

This paper is concerned with the regularity criterion for a class of axisymmetric solutions to 3D incompressible magnetohydrodynamic equations. More precisely, for the solutions that have the form of u = u r e r+ u θ e θ+ u z e z and b = b θ e θ, we prove that if | r u( x, t)|≤ C holds for −1≤ t < 0, then ( u, b) is regular at time zero. This result can be thought as a generalization of recent results in for the 3D incompressible Navier-Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the case of Prodi–Serrin type regularity criterion involving u 3 and ∂ 3 u h . More precisely, it is shown that Leray’s weak solutions of the three-dimensional Navier–Stokes equations become regular if the third component of velocity (or the gradient of the velocity field) satisfies the additional end-point Prodi–Serrin type condition. [ABSTRACT FROM AUTHOR]

In this paper, we consider the classical three-dimensional (3D) incompressible Navier–Stokes equations. By making use of a new anisotropic Sobolev inequality, we improve a regularity criterion for the weak solution to the 3D Navier–Stokes equations involving the gradient of one velocity component. This result improves the previous work of Zhou and Pokorný (2010). [ABSTRACT FROM AUTHOR]

Smooth solutions to the axially symmetric Navier–Stokes equations obey the following maximum principle: ‖ r u θ ( r , z , t ) ‖ L ∞ ≤ ‖ r u θ ( r , z , 0 ) ‖ L ∞ . We first prove the global regularity of solutions if ‖ r u θ ( r , z , 0 ) ‖ L ∞ or ‖ r u θ ( r , z , t ) ‖ L ∞ ( r ≤ r 0 ) is small compared with certain dimensionless quantity of the initial data. This result improves the one in Zhen Lei and Qi S. Zhang [10] . As a corollary, we also prove the global regularity under the assumption that | r u θ ( r , z , t ) | ≤ | ln ⁡ r | − 3 / 2 , ∀ 0 < r ≤ δ 0 ∈ ( 0 , 1 / 2 ) . [ABSTRACT FROM AUTHOR]

In this article, we study the global regularity of the 3D Navier–Stokes equations in terms of one velocity component. In particular, we establish a new version of regularity criterion in the framework of the anisotropic Lebesgue space. It is a generalization of the notable work of C. Cao and E.S. Titi (2011) [3] . [ABSTRACT FROM AUTHOR]

In this paper, we establish a new regularity criterion for the 3D Navier–Stokes equations via partial components of the fractional derivative. The main result shows that regularity criterion established on the fractional derivatives of u can be almost replaced by several of them. [ABSTRACT FROM AUTHOR]

This paper concerns the 3D Navier-Stokes equations and prove an almost Serrin-type regularity criterion in terms of one directional derivative of the pressure. [ABSTRACT FROM AUTHOR]

In this paper, we provide a regularity criterion for the Navier–Stokes equations involving the diagonal entry of the velocity gradient, which says that if ∂ 3 u 3 ∈ L ∞ ( 0 , T ; L 2 ( R 3 ) ) , then the weak solution is smooth on ( 0 , T ) . This is an important case in the sense that the strong solution is in this class. Moreover, we verify the limiting case of one regularity criterion in Cao and Titi (2011). [ABSTRACT FROM AUTHOR]