Large-time behavior of solutions to the 2D generalized magneto-micropolar equations

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Abstract

This work provides rigorous verification of the super-smoothing effect of higher-order fractional Laplacian dissipation. Shang and Zhao (2017) have proved the global regularity of classical solutions of the 2D incompressible magneto-micropolar equations with linear velocity damping u, microrotational dissipation (−∆)ω, and magnetic diffusion $(-\Delta)^{\beta} b, \beta>1$. This paper is devoted to further investigating the large-time behavior of global smooth solutions of the system with $1<\beta \leq \frac{3}{2}$. We apply the negative Sobolev space to overcome the difficulty caused by fractional-order dissipation and establish $\int_{0}^{t}\|\nabla b(\tau)\|_{L^{2}} d \tau \leq C$. Furthermore, by fully exploiting the special structure of the system and combining the properties of a heat operator with the generalized Fourier splitting methods, we obtain the decay rates of the solutions and their first-order derivatives for $1 \leq p \leq \frac{2}{\beta}$.

Keywords

magneto-micropolar equations / decay rates / fractional magnetic dissipation / linear velocity damping / Hardy-Littlewood-Sobolev inequality

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