# The Gevrey analyticity and decay for the micropolar system in the critical Besov space

The Gevrey analyticity and decay for the micropolar system in the critical Besov space In this paper, we are concerned with the 3-D incompressible micropolar fluid system, which is a non-Newtonian fluid exhibiting micro-rotational effects and micro-rotational inertia. We aim at establishing the global Gevrey analyticity in the critical Besov space. As a first step, inspired by Chemin’s work (J Anal Math 77:27–50, 1999), we construct the global-in-time existence of the strong solutions in a more general Besov space B˙p,q3p-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\dot{B}^{\frac{3}{p}-1}_{p,q}$$\end{document} with 1≤p<∞,1≤q≤∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1\le p<\infty , 1\le q\le \infty$$\end{document}. A new effective variable R=∇×ω+12Δu\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R=\nabla \times \omega +\frac{1}{2}\Delta u$$\end{document} is introduced at the low frequencies, which allows to eliminate the linear coupling terms ∇×u\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nabla \times u$$\end{document} and ∇×ω\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nabla \times \omega$$\end{document}, and obtain a global priori estimate. Secondly, observing the parabolic behaviors for u and ω\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega$$\end{document}, we would establish Gevrey analyticity based on the work by Bae, Biswas and Tadmor (Arch Ration Mech Anal 205:963–991, 2012) for the incompressible Navier–Stokes equations. The idea of effective velocity is also essential for establishing the Gevrey analyticity. As a by-product, the time-decay estimates on any derivative of solutions are also available for large time. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

# The Gevrey analyticity and decay for the micropolar system in the critical Besov space

, Volume 21 (4) – Dec 1, 2021
21 pages

/lp/springer-journals/the-gevrey-analyticity-and-decay-for-the-micropolar-system-in-the-sh840eXM0f
Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-021-00731-0
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we are concerned with the 3-D incompressible micropolar fluid system, which is a non-Newtonian fluid exhibiting micro-rotational effects and micro-rotational inertia. We aim at establishing the global Gevrey analyticity in the critical Besov space. As a first step, inspired by Chemin’s work (J Anal Math 77:27–50, 1999), we construct the global-in-time existence of the strong solutions in a more general Besov space B˙p,q3p-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\dot{B}^{\frac{3}{p}-1}_{p,q}$$\end{document} with 1≤p<∞,1≤q≤∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1\le p<\infty , 1\le q\le \infty$$\end{document}. A new effective variable R=∇×ω+12Δu\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R=\nabla \times \omega +\frac{1}{2}\Delta u$$\end{document} is introduced at the low frequencies, which allows to eliminate the linear coupling terms ∇×u\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nabla \times u$$\end{document} and ∇×ω\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nabla \times \omega$$\end{document}, and obtain a global priori estimate. Secondly, observing the parabolic behaviors for u and ω\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega$$\end{document}, we would establish Gevrey analyticity based on the work by Bae, Biswas and Tadmor (Arch Ration Mech Anal 205:963–991, 2012) for the incompressible Navier–Stokes equations. The idea of effective velocity is also essential for establishing the Gevrey analyticity. As a by-product, the time-decay estimates on any derivative of solutions are also available for large time.

### Journal

Journal of Evolution EquationsSpringer Journals

Published: Dec 1, 2021

Keywords: Critical Besov space; Micropolar system; Gevrey analyticity; 35A20; 39A14; 42B25; 46N20; 76A05

### References

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