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In this article, some elementary observations are is made regarding the behavior of solutions to the two-dimensional curl-free Burgers equation which suggests the distinguished role played by the scalar divergence field in determining the dynamics of the solution. These observations inspire a new divergence-based regularity condition for the two-dimensional Kuramoto–Sivashinsky equation (KSE) that provides conceptual clarity to the nature of the potential blow-up mechanism for this system. The relation of this regularity criterion to the Ladyzhenskaya–Prodi–Serrin-type criterion for the KSE is also established, thus providing the basis for the development of an alternative framework of regularity criterion for this equation based solely on the low-mode behavior of its solutions. The article concludes by applying these ideas to identify a conceptually simple modification of KSE that yields globally regular solutions, as well as providing a straightforward verification of this regularity criterion to establish global regularity of solutions to the 2D Burgers–Sivashinsky equation. The proofs are direct, elementary, and concise. [ABSTRACT FROM AUTHOR] more...

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 mainly proved that the weak solution is regular on (0, T] provided that either the norm π L α , ∞ (0 , T ; L β , ∞ (R 3)) with 2 α + 3 β = 2 and 3 2 < β < ∞ or ∇ π L α , ∞ (0 , T ; L β , ∞ (R 3)) with 2 α + 3 β = 3 and 1 < β < ∞ is sufficiently small. [ABSTRACT FROM AUTHOR] more...

In this paper, we prove the local well-posedness and regularity criterion of strong solutions to the 3D density-dependent incompressible Maxwell–Navier–Stokes system in a bounded and simply connected domain with vacuum. [ABSTRACT FROM AUTHOR] more...

The question of whether the two-dimensional inviscid Boussinesq equations can develop a finite-time singularity from general initial data is a challenging open problem. In this paper, we obtain two new regularity criteria for the local-in-time smooth solution to the two-dimensional inviscid Boussinesq equations. Similar result is also valid for the nonlocal perturbation of the two-dimensional incompressible Euler equations. [ABSTRACT FROM AUTHOR] more...

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] more...

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] more...

The goal of this paper is to establish a regularity criterion for the three-dimensional incompressible Hall-magnetohydrodynamic equations involving only the vorticity. Our result complements and improves some existing results. [ABSTRACT FROM AUTHOR] more...

In this paper, we give a BKM-type blow-up criterion, which involves only a single component of the velocity, for the inviscid axially symmetric Boussinesq system. More precisely, we will show that if the vorticity of the swirl part of the velocity belongs to L 1 (0 , T ∗ , L ∞) , then the solution is regular up to time T ∗ . At present, our results can not be easily generalized to the classical L t 1 B M O regularity criterion, which will be considered in our further works. [ABSTRACT FROM AUTHOR] more...

A one-component regularity criterion for the non-resistive axially symmetric Hall-MHD system is given in this paper. More precisely, we show that strong solutions to a family of axially symmetric initial data, whose velocity and current density have trivial swirl components, could be smoothly extended beyond a possible blow-up time T ∗ if and only if the swirl component of the magnetic field h θ satisfies a Beale-Kato-Majda-type criterion. See (1.5) below. This criterion is not trivial even if the velocity field vanishes since the quantity h θ r satisfies a partial Burgers' equation in this situation. [ABSTRACT FROM AUTHOR] more...

In this paper, we prove a single-component BKM-type regularity criterion for the inviscid axially symmetric Hall-MHD system. More precisely, we show that if the current density is swirl-free, then the L t 1 L x ∞ boundedness of partial vorticity (corresponding to curl of the swirl part of the velocity) implies regularity of the solution. The novelty of our result is that we only impose a critical regularity criterion on one component of the velocity which is new even if we ignore the magnetic field and consider only the 3D axially symmetric Euler equations. Our results can not be easily extended to the classical L t 1 B M O norm regularity criterion at present, which will be pursued in our further works. [ABSTRACT FROM AUTHOR] more...

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] more...

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] more...

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] more...

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] more...

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] more...

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] more...

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] more...

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] more...

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] more...

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] more...

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] more...

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] more...

In this paper, we investigate the Cauchy problem for the two-dimensional (2D) incompressible Bénard equations. On the one hand, we prove global-in-time existence of smooth solutions to the 2D Bénard equations with critical dissipation. On the other hand, we establish several regularity criteria involving temperature for 2D Bénard equations with supercritical dissipation. [ABSTRACT FROM AUTHOR] more...

This paper focuses on the global regularity problem for a family of active scalar equations with fractional dissipation. We obtain an explicit lower bound on the local existence of solutions, the eventual regularity of solutions and the global regularity of solutions for large initial data for the supercritical active scalar equations. In addition, a new regularity criterion is presented for the supercritical case which is used to study the eventual regularity. [ABSTRACT FROM AUTHOR] more...

In this paper we investigate the regularity criterion for the local-in-time classical solution to the three-dimensional (3D) incompressible Hall-magnetohydrodynamic equations (Hall-MHD). It is proved that the control of the vorticity alone can ensure the smoothness of the solution. [ABSTRACT FROM AUTHOR] more...

In this paper, we consider the axisymmetric Navier–Stokes equations with swirl. By invoking the magic identity of Miao and Zheng, the symmetry properties of Riesz transforms and the Hardy–Sobolev inequality, we establish regularity criterion involving r d ω θ with − 1 ≤ d < 0 . This improves and extends the results of [3,21]. [ABSTRACT FROM AUTHOR] more...

This paper studies the global regularity of classical solutions to 2D magneto-micropolar fluid equations with only micro-rotational velocity dissipation and magnetic diffusion. Here the micro-rotational velocity dissipation and magnetic diffusion are given by − Δ Ω and ( − Δ ) β b . Making use of several combined quantities, maximal regularity of heat operator and Littlewood–Paley decomposition theory, we establish a regularity criterion in terms of magnetic field for the case β = 1 and the global regularity for β > 1 . The regularity criterion given here is also new even for the 2D magnetohydrodynamic equations. In addition, to prove these two main results, as preparation we establish a new global a priori estimate for magnetic field, namely Δ b ∈ L ∞ ( 0 , T ; L p ( R 2 ) ) with p ≥ 2 which also holds for the 2D magnetohydrodynamic equations as a particular case. [ABSTRACT FROM AUTHOR] more...

This paper proves a regularity criterion for the 3D generalized Hall-MHD system. [ABSTRACT FROM AUTHOR]

In this paper we study the three-dimensional (3D) incompressible Hall-magnetohydrodynamic (Hall-MHD) equations and obtain its regularity criterion only in terms of the vorticity field. As a consequence, it improves the known results. [ABSTRACT FROM AUTHOR] more...

This paper studies the regularity problem for the 3D incompressible resistive viscous Hall-magneto-hydrodynamic (Hall-MHD) system. The Kolmogorov 41 phenomenological theory of turbulence [14] predicts that there exists a critical wavenumber above which the high frequency part is dominated by the dissipation term in the fluid equation. Inspired by this idea, we apply an approach of splitting the wavenumber combined with an estimate of the energy flux to obtain a new regularity criterion. The regularity condition presented here is weaker than conditions in the existing criteria (Prodi–Serrin type criteria) for the 3D Hall-MHD system. [ABSTRACT FROM AUTHOR] more...

The aim of this paper is to establish a logarithmically improved the regularity criterion in terms of the homogeneous Besov spaceto the Boussinesq equations. We prove the solutionis smooth up to timeprovided that for someand. [ABSTRACT FROM AUTHOR] more...

In this paper, we consider the density-dependent Boussinesq equations, and provide a regularity criterion involving the velocity field in BMO space. [ABSTRACT FROM AUTHOR]

In this paper, we establish a regularity criterion in terms of the temperature for the 2D fractional Boussinesq equations with zero heat conductivity. [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] more...

In this paper, we establish a new regularity criterion for the 3D incompressible MHD equations involving partial components of the velocity gradient and magnetic fields. [ABSTRACT FROM AUTHOR]

Abstract: This paper proves a logarithmically improved Beale–Kato–Majda’s criterion for the 3D nematic liquid crystal flows, which says that if then the solution is smooth up to the time . [Copyright &y& Elsevier] more...

Abstract: In this article we consider the following generalized quasi-geostrophic equation where , , and . We first show a general conditional criterion yielding the nonlocal maximum principles for the whole space active scalars, then mainly by applying the general criterion, for the case and we obtain the global well-posedness of the system with smooth initial data; and for the case and we prove the local smoothness and the eventual regularity of the weak solution of the system with appropriate initial data. [Copyright &y& Elsevier] more...

Abstract: In this paper, we consider the regularity criterion of axisymmetric weak solutions to the Navier–Stokes equations in . Let be an axisymmetric weak solution in , , and be the azimuthal component of in the cylindrical coordinates. It is proved that becomes a regular solution if , where  is the critical Morrey–Campanato space. [Copyright &y& Elsevier] more...

Abstract: In this paper, we consider regularity criterion for solutions to the 3D viscous incompressible MHD equations in Morrey–Campanato spaces. It is proved that if the vorticity field belongs to for on , then the solution remains smooth on . [Copyright &y& Elsevier] more...

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] more...

In this paper, some improved regularity criteria for the 3D magneto-micropolar fluid equations are established in Morrey–Campanato 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] . By the embedding $${ L^{\frac{3}{r}} \subset \overset{.}{\mathcal{M}}_{p,\frac{3}{r}}}$$ , we see that our result is an improvement of (Yuan in Acta Mathematica Scientia, to appear). [ABSTRACT FROM AUTHOR] more...

Abstract: We prove a regularity criterion of the strong solutions for a Bona–Colin–Lannes system in Besov space. As a byproduct, we show the existence of globally smooth solutions for a symmetric Bona–Colin–Lannes system. [Copyright &y& Elsevier] more...