Equivalence between invariant measures and statistical solutions for the 2D non-autonomous magneto-micropolar fluid equations

In this article, we aim to investigate the regularity of statistical solution for the 2D non-autonomous magneto-micropolar fluid equations as well as the relationship between invariant measures and statistical solutions. Firstly, to get the regularity of the statistical solution, we prove the existence and regularity of the pullback attractor for the equations. Then we prove the statistical solution possesses some regularity properties by using regularity of the pullback attractor. Finally, we prove the statistical solution is actual an invariant measure for the equations.

Global well-posedness and optimal time decay rates of solutions to the three-dimensional magneto-micropolar fluid equations

This paper deals with the global existence and decay estimates of solutions to the three-dimensional magneto-micropolar fluid equations with only velocity dissipation and magnetic diffusion in the whole space with various Sobolev and Besov spaces. Specifically, we first investigate the global existence and optimal decay estimates of weak solutions. Then we prove the global existence of solutions with small initial data in $H^s$, $B_{2, \infty}^s$ and critical Besov spaces, respectively. Furthermore, the optimal decay rates of these global solutions are correspondingly established in $\dot{H}^m$ and $\dot{B}_{2, \infty}^m$ spaces with $0\leq m\leq s$ and in $\dot{B}_{2, 1}^{m}$ with $0\leq m\leq \frac 12$, when the initial data belongs to $\dot{B}_{2, \infty}^{-l}$ ($0< l\leq\frac32$). The main difficulties lie in the presence of linear terms and the lack of micro-rotation velocity dissipation. To overcome them, we make full use of the special structure of the system and employ various techniques involved with the energy methods, the improved Fourier splitting, Fourier analysis and the regularity interpolation methods.