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Global strong solutions for the incompressible micropolar fluids equations

  • Felipe W. CruzEmail author
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Abstract

We study the Cauchy problem for the system of equations describing the nonstationary flow of micropolar incompressible fluids in \(\mathbb {R}^3\). We show the global existence and uniqueness of strong solutions with initial velocities \({\varvec{u}}_{0}, \,{\varvec{w}}_{0} \in {\varvec{H}}^1 (\mathbb {R}^3)\) satisfying
$$\begin{aligned} \left( \bigl \Vert {\varvec{u}}_{0}\bigr \Vert ^{2}_{{\varvec{L}}^2} + \bigl \Vert {\varvec{w}}_{0} \bigr \Vert ^{2}_{{\varvec{L}}^2}\right) \! \times \! \left( \bigl \Vert \nabla {\varvec{u}}_0 \bigr \Vert ^{2}_{{\varvec{L}}^2} + \bigl \Vert \nabla {\varvec{w}}_0 \bigr \Vert ^{2}_{{\varvec{L}}^2} + \bigl \Vert {\mathrm{curl}}\,{\varvec{u}}_0 - 2 {\varvec{w}}_0 \bigr \Vert ^{2}_{{\varvec{L}}^2}\right) \end{aligned}$$
being sufficiently small.

Keywords

Global regularity Micropolar fluids Strong solutions 

Mathematics Subject Classification

35Q30 76D03 35Q35 

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Notes

Acknowledgements

The author would like to thank Marko A. Rojas-Medar for helpful comments and fruitful discussions on this paper.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal de PernambucoRecifeBrazil

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