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This paper mainly investigates the large time behavior of the magneto-micropolar equations with zero angular viscosity in R d (d = 2, 3). The main difficulties lie in the presence of linear curl terms and the lack of micro-rotation velocity dissipation. By utilizing the effect of velocity dissipation and damping term in the equation of micro-rotational velocity, we prove the non-uniform decay and the upper bounds of decay estimates of the global solutions. Furthermore, the lower bounds of decay estimates are shown by making full use of the structure of this system and by introducing a combined quantity. The approach used here can also be applied to derive the lower and upper bounds of decay for the magneto-micropolar equations with full dissipation. We remark that the lower bounds are new even for the fully dissipative system. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the global well-posedness of solutions to three-dimensional incompressible Boussinesq equations with temperature-dependent viscosity under the smallness assumption of initial velocity fields u0 in the critical space B ̇ 3 , 1 0 . The key ingredients here lie in the decomposition of the velocity fields and the regularity theory of the Stokes system, which are crucial to get rid of the smallness restricition of the initial temperature θ0. In addition, we mention that the improved decay estimates in time is also necessary. [ABSTRACT FROM AUTHOR]

In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flows of viscous incompressible magnetohydrodynamic flow with the no-slip boundary condition of velocity and perfectly conducting walls for magnetic fields. The convergence is shown under various Sobolev norms, including the physically important space–time uniform norm L∞(H1). In addition, similar convergence results are also obtained under the case with uniform magnetic fields. This implies the stabilizing effects of magnetic fields. Besides, the higher-order expansion is also considered. [ABSTRACT FROM AUTHOR]