Micropolar fluids with vanishing viscosity. (English) Zbl 1417.76006

Summary: A study of the convergence of weak solutions of the nonstationary micropolar fluids, in bounded domains of \(\mathbb R^{n}\), when the viscosities tend to zero, is established. In the limit, a fluid governed by an Euler-like system is found.


76A05 Non-Newtonian fluids
35Q30 Navier-Stokes equations
Full Text: DOI


[1] A. C. Eringen, “Theory of micropolar fluids,” Journal of Mathematics and Mechanics, vol. 16, pp. 1-18, 1966. · Zbl 0145.21302
[2] R. Temam, Navier-Stokes Equations, vol. 2, North-Holland, Amsterdam, The Netherlands, 2nd edition, 1979. · Zbl 0426.35003
[3] D. W. Condiff and J. S. Dahler, “Fluid mechanical aspects of antisymmetric stress,” The Physics of Fluids, vol. 7, pp. 842-854, 1964. · Zbl 0125.15801 · doi:10.1063/1.1711295
[4] G. Lukaszewicz, Micropolar Fluids. Theory and application, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, Mass, USA, 1999. · Zbl 0923.76003
[5] M. A. Rojas-Medar and E. E. Ortega-Torres, “The equations of a viscous asymmetric fluid: an interactive approach,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 85, no. 7, pp. 471-489, 2005. · Zbl 1072.35568 · doi:10.1002/zamm.199910189
[6] M. A. Rojas-Medar and J. L. Boldrini, “Magneto-micropolar fluid motion: existence of weak solutions,” Revista Matemática de la Universidad Complutense de Madrid, vol. 11, no. 2, pp. 443-460, 1998. · Zbl 0918.35114 · doi:10.5209/rev_REMA.1998.v11.n2.17276
[7] L. C. F. Ferreira and E. J. Villamizar-Roa, “On the existence and stability of solutions for the micropolar fluids in exterior domains,” Mathematical Methods in the Applied Sciences, vol. 30, no. 10, pp. 1185-1208, 2007. · Zbl 1117.35065 · doi:10.1002/mma.838
[8] E. J. Villamizar-Roa and M. A. Rodríguez-Bellido, “Global existence and exponential stability for the micropolar fluid system,” Zeitschrift für Angewandte Mathematik und Physik, vol. 59, no. 5, pp. 790-809, 2008. · Zbl 1155.76009 · doi:10.1007/s00033-007-6090-2
[9] P. Braz e Silva, E. Fernández-Cara, and M. A. Rojas-Medar, “Vanishing viscosity for non-homogeneous asymmetric fluids in R3,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 833-845, 2007. · Zbl 1121.35105 · doi:10.1016/j.jmaa.2006.10.066
[10] B. A. Ton, “Nonstationary Navier-Stokes flows with vanishing viscosity,” Rendiconti del Circolo Matematico di Palermo, vol. 27, no. 2, pp. 113-129, 1978. · Zbl 0446.76038 · doi:10.1007/BF02843932
[11] D. G. Ebin and J. E. Marsden, “Groups of diffeomorphisms and the motion of an incompressible fluid,” Annals of Mathematics, vol. 92, pp. 102-163, 1970. · Zbl 0211.57401 · doi:10.2307/1970699
[12] T. Kato, “Non stationary flows of viscous and ideal fluids in R3,” Journal of Functional Analysis, vol. 9, pp. 296-305, 1972. · Zbl 0229.76018 · doi:10.1016/0022-1236(72)90003-1
[13] H. S. Swann, “The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in R3,” Transactions of the American Mathematical Society, vol. 157, pp. 373-397, 1971. · Zbl 0218.76023 · doi:10.2307/1995853
[14] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires,, Dunod, Paris, France, 1969. · Zbl 0189.40603
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.