On the strong solutions of the 3D magneto-micropolar equations

ABSTRACT

This paper is dedicated to the study of micropolar fluids which is moving in the presence of a magnetic field in ${\mathbb{R}}^3$. We prove the global existence and uniqueness of strong solutions with initial data $\left({\mathit{u}}_0,{\mathit{w}}_0,{\mathit{b}}_0\right)\in {\mathit{H}}_{\sigma }^1\left({\mathbb{R}}^3\right)\times {\mathit{H}}^1\left({\mathbb{R}}^3\right)\times {\mathit{H}}_{\sigma }^1\left({\mathbb{R}}^3\right)$ satisfying $\begin{array}{rl}& \left(\parallel {\mathit{u}}_0{\parallel }_{{\mathit{L}}^2}^2+\parallel {\mathit{w}}_0{\parallel }_{{\mathit{L}}^2}^2+\parallel {\mathit{b}}_0{\parallel }_{{\mathit{L}}^2}^2\right)\\ & \times \left(\parallel \mathrm{\nabla }{\mathit{u}}_0{\parallel }_{{\mathit{L}}^2}^2+\parallel \mathrm{\nabla }{\mathit{w}}_0{\parallel }_{{\mathit{L}}^2}^2+\parallel \mathrm{\nabla }{\mathit{b}}_0{\parallel }_{{\mathit{L}}^2}^2+\parallel \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\mathit{u}}_0-2{\mathit{w}}_0{\parallel }_{{\mathit{L}}^2}^2\right)\end{array}$ being small enough.

Keywords:

• Magneto-micropolar fluids
• strong solutions
• regularity theory

2010 Mathematics Subject Classifications:

• 35Q30
• 35Q35
• 76D03

Acknowledgments

Felipe W. Cruz was partially supported by (CAPES) – PRINT, #88881.311964/2018–01 and Propesq-UFPE, Editais #08-2019 (Qualis A) and #09-2019 (Produtividade em Pesquisa).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Felipe W. Cruz was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) – PRINT, #88881.311964/2018–01 and Pró-Reitoria de Pesquisa, Universidade Federal de Pernambuco (Propesq-UFPE, Editais) #08-2019 (Qualis A) and #09-2019 (Produtividade em Pesquisa).