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This paper is concerned with the logarithmically improved regularity criterion in terms of the middle eigenvalue of the strain tensor to the 3D Boussinesq equations in Besov spaces with negative indices. It is shown that a weak solution is regular on $ (0, T] $ provided that$ \begin{align*} \int_{0}^{T}\frac{\left\Vert \lambda _{2}^{+}(\cdot , t)\right\Vert _{\dot{B} _{\infty , \infty }^{-\delta }}^{\frac{2}{2-\delta }}}{\ln (e+\left\Vert u(\cdot , t)\right\Vert _{\dot{B}_{\infty , \infty }^{-\delta }})}dt<\infty. \end{align*} $for some $ 0<\delta <1 $. As a consequence, this result is some improvements of recent works [11,12] established by Neustupa-Penel and Miller. [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.

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 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]

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 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

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 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]

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]

In this short paper, we consider the two-dimensional inviscid Boussinesq equations and establish an improved regularity criterion. [ABSTRACT FROM AUTHOR]

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]

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]

This paper proves some regularity criteria for some simplified non-isothermal models for nematic liquid crystals. [ABSTRACT FROM AUTHOR]

In this paper, we consider the two-dimensional (2D) Boussinesq equations with partial dissipation. The issue of whether the 2D Boussinesq equations always possess global (in time) classical solutions can be difficult when there is only partial dissipation. Therefore, the goal of this paper is to establish some regularity criteria of the corresponding systems. [ABSTRACT FROM AUTHOR]

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]

In this paper, we prove the following regularity criterion ω : = curl u ∈ L 1 ( 1 , T ; B M O ) for the 3D Boussinesq system with zero heat conductivity in a bounded domain. Here ω is the vorticity of the fluid velocity field u and BMO denotes the function space of bounded mean oscillation. [ABSTRACT FROM AUTHOR]

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 establish the global weak solutions for two-dimensional Boussinesq equations with mixed partial dissipation and thermal diffusivity. Moreover, we give global regularity criterion for weak solutions to the two-dimensional Boussinesq equations with mixed partial dissipation and thermal diffusivity. [ABSTRACT FROM AUTHOR]

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 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]

Abstract: In this note, we consider the regularity problem under the critical condition to the Boussinesq equations with zero heat conductivity. The Serrin type regularity criteria are established in terms of the critical Besov spaces. This improves a result established in a recent work by Geng and Fan (2012) [6]. [Copyright &y& Elsevier]

In this paper, we consider the three-dimensional (3D) incompressible Boussinesq equations. We obtain some regularity criteria for the three-dimensional (3D) Boussinesq equations in the Morrey-Campanato space. [ABSTRACT FROM AUTHOR]

This note is devoted to the study of the regularity of strong solutions to the Boussinesq system in which the vorticity and to improve a result of Fan and Ozawa [J. Fan, T. Ozawa, Regularity conditions for the 3D Boussinesq equations with partial viscosity terms, Nonlinearity 22 (2009), pp. 553–568] and Gala [S. Gala, Remark on the regularity criterion on strong solutions to the 3D Boussinesq equations (Submitted)]. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we consider the 3D Boussinesq equations with the incompressibility condition. We obtain a regularity condition for the three-dimensional Boussinesq equations by means of the Littlewood–Paley theory and Bony’s paradifferential calculus. [Copyright &y& Elsevier]

Abstract: In this note we prove a regularity criterion for the 3D Boussinesq system with partial viscosity. Here is the velocity, is the vorticity and denotes the homogeneous Besov space. [Copyright &y& Elsevier]

In this paper we establish a new regularity criterion for the 3D incompressible MHD equations via ∂ 3 u 3 and the magnetic field b. By considering different weights in spatial variables, we show in anisotropic Lebesgue spaces if ∂ 3 u 3 and b satisfy certain space–time integrable conditions, which are almost optimal from the scaling invariant point of view, then a weak solution (u , b) is actually regular. This result gives new insights into the understanding of regularity theory of weak solutions. [ABSTRACT FROM AUTHOR]

In this paper, we provide a regularity criterion for 3D Navier–Stokes equations in terms of two vorticity components, which extends a recent result established by Guo, Kučera and Skalák (2018). More precisely, we prove that a unique local strong solution u to 3D Navier–Stokes equations does not blow up at time T provided only two components of vorticity belongs to L 2 (0 , T ; B M O − 1). We also prove that u can be smoothly extended beyond T , if the horizontal gradient of horizontal components of velocity u belong to L 2 (0 , T ; B M O − 1). The proof relies on the technique of the Bony decomposition and some Fourier multiplier theorems. [ABSTRACT FROM AUTHOR]