Search

Abstract 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 ) $(u,\omega,b)$ to the magneto-micropolar fluid flows can be extended beyond time t = T $t=T$ , provided if u 3 ∈ L β ( 0 , T ; L α ( R 3 ) ) $u_{3}\in L^{\beta }(0,T;L^{\alpha }(R^{3}))$ with 2 β + 3 α ≤ 3 4 + 1 2 α , α > 10 3 $\frac{2}{\beta }+\frac{3}{\alpha }\leq \frac{3}{4}+\frac{1}{2\alpha },\alpha > \frac{10}{3}$ and ∇ b ∈ L 4 p 3 ( p − 2 ) ( 0 , T ; M ˙ p , q ( R 3 ) ) $\nabla b\in L^{\frac{4p}{3(p-2)}}(0,T;\dot{M}_{p,q}(R^{3}))$ with 1 < q ≤ p < ∞ $1< q\leq p

Abstract 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 ) $L^{\infty }(0,T;\dot{B}_{\infty,\infty }^{-1})$ -norm of a suitable low frequency part of ( u , ∇ d ) $(u,\nabla d)$ is bounded by a scaling invariant constant and the initial data ( u 0 , ∇ d 0 ) $(u_{0},\nabla d_{0})$ . 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).

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]

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 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]

Abstract We consider a Keller-Segel-Euler model and prove a regularity criterion of the local strong solutions in a 3D bounded domain Ω.

We consider a Keller-Segel-Euler model and prove a regularity criterion of the local strong solutions in a 3D bounded domain Ω. [ABSTRACT FROM AUTHOR]

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

In this paper we consider an initial value problem of the 2D MHD equations with velocity dissipation and without magnetic diffusion. We establish a new magnetic regularity criterion in terms of the magnetic field. In contrast to the magnetic regularity criterion $\nabla b\in L^{1}(0,T; BMO(\mathbb {R}^{2}))$, our regularity criterion $\int_{0}^{T} (\Vert b\otimes b(s)\Vert _{B_{\infty,1}^{0}(\mathbb {R}^{2})} +\Vert b\otimes b(s)\Vert _{L^{2}(\mathbb {R}^{2})} )\,ds<\infty$ is different; for example, our simplified regularity criterion $\int_{0}^{T}\Vert b(s)\Vert ^{2}_{B_{\infty,1}^{\varepsilon}(\mathbb {R}^{2})}\,ds<\infty$ requires higher time integrability and lower regularity of space. [ABSTRACT FROM AUTHOR]

In this paper, we consider the three-dimensional generalized MHD equations, a system of equations resulting from replacing the Laplacian -δ in the usual MHD equations by a fractional Laplacian (δ)ɑ. We obtain a regularity criterion of the solution for the generalized MHD equations in terms of the summation of the velocity field u and the magnetic field b by means of the Littlewood-Paley theory and the Bony paradifferential calculus, which extends the previous result. [ABSTRACT FROM AUTHOR]

This paper concentrates on the global regularity of classical solution to the 212D 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 212D 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 212D 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 212D 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∞-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). [ABSTRACT FROM AUTHOR]