Existence of global strong solution to the micropolar fluid system in a bounded domain. (English) Zbl 1078.35096

Summary: We are concerned with the initial boundary value problem of the micropolar fluid system in a three dimensional bounded domain. We study the resolvent problem of the linearized equations and prove the generation of analytic semigroup and its time decay estimates. In particular, \(L^{p}\)-\(L^{q}\) type estimates are obtained. By use of the \(L^{p}\)-\(L^{q}\) estimates for the semigroup, we prove the existence theorem of global in time solution to the original nonlinear problem for small initial data. Furthermore, we study the magneto-micropolar fluid system in the final section.


35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76A05 Non-Newtonian fluids
76W05 Magnetohydrodynamics and electrohydrodynamics
Full Text: DOI


[1] Eringen, Journal of Mathematical Mechanics 16 pp 1– (1966)
[2] Galdi, International Journal of Engineering Science 15 pp 105– (1977)
[3] Lukaszewicz, Rendiconti Accademia Nazionale delle Scienze della dei XL Serie V. Memorie di Matematica 12 pp 83– (1988)
[4] Kagei, Hiroshima Mathematical Journal 23 pp 343– (1993)
[5] Inoue, Discrete and Continuous Dynamical Systems pp 439– (2003)
[6] Fujita, Archives for Rational Mechanics and Analysis 16 pp 269– (1964)
[7] Giga, Archives for Rational Mechanics and Analysis 89 pp 267– (1985)
[8] Kato, Mathematische Zeitschrift 187 pp 471– (1984)
[9] Farwig, Journal of Mathematical Society of Japan 46 pp 607– (1994)
[10] Fujiwara, Journal of the Faculty of Science, University of Tokyo, Section IA, Mathematics 24 pp 685– (1977)
[11] . An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vol. I. Linearized Steady Problems. Springer: New York, 1994. · Zbl 0949.35004
[12] Agmon, Communications on Pure and Applied Mathematics 12 pp 623– (1959)
[13] Agmon, Communications on Pure and Applied Mathematics 17 pp 35– (1964)
[14] . Theory of Function Spaces. II, Birkhäuser, Basel, 1992. · Zbl 1235.46003 · doi:10.1007/978-3-0346-0419-2
[15] Yoshida, Journal of Mathematical Physics 24 pp 2860– (1983)
[16] . On an existence theorem of global strong solution to the magnetohydrodynamic system in three dimensional exterior domain, preprint.
[17] Rojas-Medar, Mathematische Nachrichten 188 pp 301– (1997)
[18] Ortega-Torres, Abstract and Applied Analysis 4 pp 109– (1999)
[19] Miyakawa, Hiroshima Mathematical Journal 10 pp 517– (1980)
[20] Akiyama, Funkcialaj Ekvacioj 47 pp 361– (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.