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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.

MSC:

76A05 Non-Newtonian fluids
35Q30 Navier-Stokes equations
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[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
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