Time asymptotic profiles to the magneto-micropolar system

ABSTRACT

This work examines some decay properties of solutions to the Cauchy problem for the magneto-micropolar fluid equations on ${\dot{H}}^s\left({\mathbb{R}}^n\right)$ (where $n=2,3$ and $s\ge 0$ real). We show, for each $s\ge 0$, that $t^{s/2}\parallel (\mathit{u},\mathit{w},\mathit{b})(\cdot ,t){\parallel }_{{\dot{H}}^s\left({\mathbb{R}}^n\right)}\to 0$ as $t\to \mathrm{\infty }$ for Leray global solutions with an arbitrary initial data in $L^2\left({\mathbb{R}}^n\right)$. When the vortex viscosity is present, we obtain a (faster) decay for the micro-rotational field: $\parallel \mathit{w}(\cdot ,t){\parallel }_{{\dot{H}}^s\left({\mathbb{R}}^n\right)}=o\left(t^{\left(s+1\right)/2}\right)$. Some related results are also included.

COMMUNICATED BY:

• Rolando Magnanini

KEYWORDS:

• Magneto-micropolar fluid equations
• long-time behavior
• decay estimates in $H^s$

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35Q35
• 35B40
• 76D05

Acknowledgements

We would like to express our gratitude to Prof. P. Zingano for his interest in this work and various helpful suggestions. We also thank the Brazilian agencies CAPES and CNPq for their financial support to this work.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. For example, we actually have ${\displaystyle \parallel \text{u}{\parallel }_{L^{\mathrm{\infty }}\left({\mathbb{R}}^2\right)}\le K\parallel \text{u}{\parallel }_{L^2\left({\mathbb{R}}^2\right)}^{1/2}\parallel D\text{u}{\parallel }_{L^2\left({\mathbb{R}}^2\right)}^{1/2}}$ with $K=1/2$, and so on. All interpolation inequalities we use have constants $K\le 1$, and so we simply take $K=1$ throughout, except where indicated otherwise.

2. Because a monotonic function $f\in C^0\left(\right(a,\mathrm{\infty }\left)\right)\cap L^1\left(\right(a,\mathrm{\infty }\left)\right)$ has to satisfy $f\left(t\right)=o\left(1/t\right)$ as $t\to \mathrm{\infty }$ (see e.g. [20, p.236]).

3. However, after we show (3a(3a) $\underset{t\to \mathrm{\infty }}{lim}{t}^{s/2}\parallel (\mathit{u},\mathit{w},\mathit{b})(\cdot ,t){\parallel }_{{\dot{H}}^s\left({\mathbb{R}}^n\right)}=0.$(3a) ) for $s=1$ it can be done just repeating the same steps to obtain (12(12) $\begin{array}{rl}& \parallel D\mathit{z}(\cdot ,t){\parallel }_{L^2\left({\mathbb{R}}^3\right)}^2+2\lambda {\int }_{t_0}^t\parallel D^2\mathit{z}(\cdot ,\tau ){\parallel }_{L^2\left({\mathbb{R}}^3\right)}^2\mathrm{d}\tau \\ & +2{\int }_{t_0}^t\parallel D\zeta (\cdot ,\tau ){\parallel }_{L^2\left({\mathbb{R}}^3\right)}^2\mathrm{d}\tau +2\chi {\int }_{t_0}^t\parallel D\mathit{w}(\cdot ,\tau ){\parallel }_{L^2\left({\mathbb{R}}^3\right)}^2\mathrm{d}\tau \le \parallel D\mathit{z}(\cdot ,t_0){\parallel }_{L^2\left({\mathbb{R}}^3\right)}^2\\ & +C{\int }_{t_0}^t\parallel \mathit{z}(\cdot ,\tau ){\parallel }_{L^2\left({\mathbb{R}}^3\right)}^{1/2}\parallel D\mathit{z}(\cdot ,\tau ){\parallel }_{L^2\left({\mathbb{R}}^3\right)}^{1/2}\parallel D^2\mathit{z}(\cdot ,\tau ){\parallel }_{L^2\left({\mathbb{R}}^3\right)}^2\mathrm{d}\tau ,\end{array}$(12) ).

4. Observe that, if $f\in L^2\left({\mathbb{R}}^n\right)\cap \dot{\phantom{a}}{H}^{s_1}\left({\mathbb{R}}^n\right)$, for $s_1>0$, then $f\in \dot{\phantom{a}}{H}^s\left({\mathbb{R}}^n\right)$, for each $0<s<s_1$ and $\parallel f{\parallel }_{{\dot{H}}^s\left({\mathbb{R}}^n\right)}\le \parallel f{\parallel }_{L^2\left({\mathbb{R}}^n\right)}^{1-\left(s/\right(s_1\left)\right)}\parallel f{\parallel }_{{\dot{H}}^{s_1}\left({\mathbb{R}}^n\right)}^{s/\left(s_1\right)}.$

Additional information

Funding

The work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grant 140448/2017-9 and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) grants 1510610, 1442127.