Hostname: page-component-7b9c58cd5d-nzzs5 Total loading time: 0 Render date: 2025-03-13T11:26:42.372Z Has data issue: false hasContentIssue false
  • English
  • Français

Well-posedness for the 3-D generalized micropolar system in critical Fourier–Besov–Morrey spaces

Part of: Partial differential equations Equations of mathematical physics and other areas of application Incompressible viscous fluids

Published online by Cambridge University Press:  26 February 2025

Peng Gao ,
Baoquan Yuan Open the ORCID record for Baoquan Yuan [Opens in a new window]  and
Tiantian Zhai
Peng Gao
Affiliation:
School of Mathematics and Information Science, Henan Polytechnic University, Henan, 454000, China e-mail: [email protected] [email protected]
Baoquan Yuan*
Affiliation:
School of Mathematics and Information Science, Henan Polytechnic University, Henan, 454000, China e-mail: [email protected] [email protected]
Tiantian Zhai
Affiliation:
School of Mathematics and Information Science, Henan Polytechnic University, Henan, 454000, China e-mail: [email protected] [email protected]
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we focus on the Cauchy problem of the three-dimensional generalized incompressible micropolar system in critical Fourier–Besov–Morrey spaces. By using the Fourier localization argument and the Littlewood–Paley theory, we get the local well-posedness results and global well-posedness results with small initial data belonging to the critical Fourier–Besov–Morrey spaces.

Keywords

Generalized micropolar systemglobal well-posednessFourier–Besov–Morrey space

MSC classification

Primary: 35B65: Smoothness and regularity of solutions 35Q35: PDEs in connection with fluid mechanics 76D03: Existence, uniqueness, and regularity theory
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 17
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bony, J. M., Calcul symbolique et propagation des singularitiés pour les équations aux dérivées partielles non linéaires . Ann. Sci. Ecole Norm. Sup. 14(1981), 209–246.CrossRefGoogle Scholar
Chen, Q. and Miao, C., Global well-posedness for the micropolar fluid system in critical Besov spaces . J. Differ. Equ. 252(2012), no. 3, 2698–2724.CrossRefGoogle Scholar
El Baraka, A. and Toumlilin, M., Global well-posedness and decay results for 3D generalized magneto-hydrodynamic equations in critical Fourier-Besov-Morrey spaces . Electron. J. Differ. Equ. 2017(2017), no. 65, 1–20.Google Scholar
El Baraka, A. and Toumlilin, M., Global well-posedness for fractional Navier-Stokes equations in critical fourier-Besov-Morrey spaces . Moroccan J. Pure Appl. Anal. 3(2017), no. 1, 1–13.CrossRefGoogle Scholar
El Baraka, A. and Toumlilin, M., Uniform well-posedness and stability for fractional Navier-Stokes equations with Coriolis force in critical Fourier-Besov-Morrey spaces , Open J. Math. Anal. 3(2019), no. 1, 70–89.CrossRefGoogle Scholar
Eringen, A. C., Theory of micropolar fluids . J Math Mech. 16(1966), 1–18.Google Scholar
Ferreira, L. C. and Lima, L. S., Self-similar solutions for active scalar equations in Fourier-Besov-Morrey spaces . Monatsh. Math. 175(2014), no. 4, 491–509.Google Scholar
Ferreira, L. C. F. and Villamizar-Roa, E. J., Micropolar fluid system in a space of distributions and large time behavior J Math Anal Appl. 332(2007), no. 2, 1425–1445.CrossRefGoogle Scholar
Galdi, G. P. and Rionero, S., A note on the existence and uniqueness of solutions of the micropolar fluid equations , Int. J. Eng. Sci. 15(1977), no. 2, 105–108.CrossRefGoogle Scholar
Kato, T., Strong ${L}^p$ -solutions of the Navier-Stokes equations in ${\mathbb{R}}^m$ with applications to weak solutions . Math. Z. 187(1984), 471–480.CrossRefGoogle Scholar
Kato, T., Strong solutions of the Navier-Stokes equation in Morrey spaces . Bull. Braz. Math. Soc. 22(1992), no. 2, 127–155.CrossRefGoogle Scholar
Lukaszewicz, G., Micropolar fluids: Theory and applications, modeling and simulation in science, engieering and technology. Birkhäuser, Boston, 1999.CrossRefGoogle Scholar
Lukaszewicz, G., Asymptotic behavior of micropolar fluid flows, The Erigen anniversary issue (University Park, PA, 2002) . J Engrg Sci 41(2003), nos. 3–5, 259–269.CrossRefGoogle Scholar
Migun, N. P. and Prohorenko, P. P., Hydrodynamics and heat exchange for gradient flows of fluid with microstructure. Naukai Technika, Minsk, 1984. (in Russian).Google Scholar
Popel, A. S., On the hydrodynamics of suspensions . Izv AN SSSR 4(1969), 24–30. (in Russian).Google Scholar
Popel, A. S., Regirer, S. A., and Usick, P. I., A continuum model of blood flow . Biorheology 11(1974), 427–437.Google ScholarPubMed
Taylor, M. E., Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations . Commun. Partial Differ. Equ. 17(1992), nos. 9–10, 1407–1456.CrossRefGoogle Scholar
Triebel, H., Theory of function spaces, Monographs in Mathematics, 78, Birkhauser Verlag, Basel, 1983.CrossRefGoogle Scholar
Xiao, W., Chen, J., Fan, D., and Zhou, X., Global well-posedness and long time decay of fractional Navier–Stokes equations in Fourier–Besov spaces . Abstr. Appl. Anal. 2014(2014), Article ID 463639, 11 pages.CrossRefGoogle Scholar
Yuan, B. Q. and Ma, L., Existence of strong solution to the Magneto-micropolar fluid equations in critical Sobolev space . Acta Math. Appl. Sin. 39(2016), no. 5, 710–718.Google Scholar
Zhao, J., Liu, Q., and Cui, S., Existence of solutions for the Debye-Hückel system with low regularity initial data . Acta Appl. Math. 125(2013), no. 1, 1–10.CrossRefGoogle Scholar