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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]

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]

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]

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]

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]

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]

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]

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 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]

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 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]

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]

We consider the Cauchy problem for the three-dimensional Navier-Stokes equations, and provide an optimal regularity criterion in terms of u3 and ω3, which are the third components of the velocity and vorticity, respectively. This gives an affirmative answer to an open problem in the paper by P. Penel, M.Pokorný (2004). [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]

In this paper, we are concerned with the regularity criterion for weak solutions to the 3D incompressible MHD equations. We show that if any two groups functions of $$(\partial _{1}u_{1}, \partial _{1}b_{1})$$ , $$(\partial _{2}u_{2}, \partial _{2}b_{2})$$ and $$(\partial _{3}u_{3}, \partial _{3}b_{3})$$ belong to the space $$L^{\theta }([0,T);L^{r}(\mathbb {R}^3)), \frac{2}{\theta }+\frac{3}{r}=2, \frac{3}{2}

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]

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]

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]

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]

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]

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

In this paper, we consider the three-dimensional Boussinesq equations, and obtain a logarithmically improved regularity criterion in terms of pressure. [ABSTRACT FROM AUTHOR]