A Note on Div-Curl Lemma

2000 Mathematics Subject Classification: 42B30, 46E35, 35B65.We prove two results concerning the div-curl lemma without assuming any sort of exact cancellation, namely the divergence and curl need not be zero, and $$div(u^−v^→) ∈ H^1(R^d)$$ which include as a particular case, the result of [3]

A new regularity criterion for the Navier-Stokes equations in terms of the two components of the velocity

This paper establishes a new regularity criterion for the Navier-Stokes equation in terms of two velocity components. We show that if the two velocity components $\widetilde{u}=\left( u_{1},u_{2},0\right)$ satisfy \begin{equation*} \int_{0}^{T}\Vert \tilde{u}(s)\Vert _{\dot{B}_{\infty ,\infty }^{0}}^{2}ds<\infty , \end{equation*} then the solution can be smoothly extended after $t=T$. This gives an aswer to an open problem in [B. Q. Dong, Z. Zhang, Nonlinear Anal. Real World Appl. 11(2010), 2415-2421]

Logarithmically improved regularity criterion for the 3D Hall-MHD equations

In this work, we study the blow-up criterion of the smooth solutions of three-dimensional incompressible Hall-magnetohydrohynamics equations (in short, Hall-MHD). We obtain a logarithmically improved regularity criterion of smooth solutions in terms of the B˙∞,∞0 norm. We improve the blow-up criterion for smooth solutions established in Ye (Appl Anal 96:2669–2683, 2016). © 2021, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional

On the blow-up criterion of strong solutions for the MHD equations with the Hall and ion-slip effects in {\mathbb{R}^{3}}

In this paper, we establish a blow-up criterion of strong solutions to the 3D incompressible magnetohydrodynamics equations including two nonlinear extra terms: the Hall term (quadratic with respect to the magnetic field) and the ion-slip term (cubic with respect to the magnetic field). This is an improvement of the recent results given by Fan et al. (Z Angew Math Phys, 2015)

PAC Fields over Finitely Generated Fields

We prove the following theorem for a finitely generated field $K$: Let $M$ be a Galois extension of $K$ which is not separably closed. Then $M$ is not PAC over $K$.Comment: 7 pages, Math.

A new regularity criterion of weak solutions to the 3D micropolar fluid flows in terms of the pressure

In this study, we establish a new regularity criterion of weak solutions to the three-dimensional micropolar fluid flows by imposing a critical growth condition on the pressure field. © 2020, Unione Matematica Italiana

On the regularity of weak solutions of the boussinesq equations in besov spaces

The main issue addressed in this paper concerns an extension of a result by Z. Zhang who proved, in the context of the homogeneous Besov space

A regularity criterion in multiplier spaces to Navier-Stokes equations via the gradient of one velocity component

In this paper, we study regularity of weak solutions to the incompressible Navier-Stokes equations in $\mathbb{R}^{3}\times (0,T)$. The main goal is to establish the regularity criterion via the gradient of one velocity component in multiplier spaces.Comment: 9 pages. arXiv admin note: text overlap with arXiv:2005.1401