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This paper is concerned with the global smooth solution of the 3D generalized magneto‐micropolar equations. When the velocity dissipation is logarithmically hyperdissipative and the magnetic diffusion is fractional Laplacian, based on some new observations on the nonlinear structure for the magneto‐micropolar equations, it is examined the system has a unique global smooth solution (u,w,b)∈C([0,T];Hs(ℝ3))$$ \left(u,w,b\right)\in C\left(\left[0,T\right];{H}^s\left({\mathrm{\mathbb{R}}}^3\right)\right) $$. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global regularity of the two-dimensional generalized Boussinesq equations. When the dissipation mechanism is only fractional vertical dissipation in the horizontal velocity and horizontal dissipation in the vertical velocity, it is examined that the system has a unique global smooth solution. The finding not only settles a problem remarked in (J. Differential Equations 268: 910–944, 2020), but also relaxes the restriction on the initial data. [ABSTRACT FROM AUTHOR]

This paper is devoted to understanding the global stability of perturbations near a background magnetic field of the 2D magnetohydrodynamic (MHD) equations with partial dissipation. We establish the global stability for the solutions of the nonlinear MHD system by the bootstrap argument. [ABSTRACT FROM AUTHOR]

This study is concerned with the large time decay rates of the H1 weak solutions for the 2D magnetohydrodynamic equations with linear damping velocity. Due to the inconsistencies of the dissipative effects between linear velocity and Laplacian magnetic diffusion, neither the classic Fourier splitting method nor Kato's method is available directly. By developing an alternative analysis argument together with an iterative process, we give a detailed description on the sharp upper decay rates for the velocity field and for the magnetic field ,. Additionally, the optimal lower decay rates of the magnetic field are also presented. [ABSTRACT FROM AUTHOR]

This paper is devoted to the global regularity for the Cauchy problem of the two-dimensional Boussinesq equations with the temperature-dependent viscosity. We prove the global solutions for this system with any positive power of the fractional Laplacian for temperature under the assumption that the viscosity coefficient is sufficiently close to some positive constant. Our obtained result improves considerably the recent results in Abidi and Zhang (Adv Math 305:1202–1249, 2017) and Zhai et al. (J Differ Equ 267:364–387, 2019). In addition, a regularity criterion via the velocity is also obtained for this system without the above assumption on the viscosity coefficient. [ABSTRACT FROM AUTHOR]