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Dong, Bo-Qing; Chen, Zhi-Min

On upper and lower bounds of higher order derivatives for solutions to the 2D micropolar fluid equations. (English) Zbl 1158.35074

J. Math. Anal. Appl. 334, No. 2, 1386-1399 (2007).
The purpose of this paper is to show the upper-lower bound estimate for the \(L^2\)-decay rates of higher-order derivatives of solutions to the micropolar fluid motion equations. To this end, the authors examine the decay estimates of derivatives for the solutions to the linearized micropolar fluid motion equations and then extend the estimates on linearized equations to the nonlinear equations by using a generalized Gronwall type argument.
Reviewer: Messoud A. Efendiev (Berlin)

Cited in 16 Documents

MSC:

35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
76A05 Non-Newtonian fluids
76D05 Navier-Stokes equations for incompressible viscous fluids

Keywords:

micropolar fluid motion equations; upper and lower bounds; \(L^{2}\) decay
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References:

[1] Borchers, W.; Miyakawa, T., \(L^2\) decay rate for the Navier-Stokes flow in half spaces, Math. Ann., 282, 139-155 (1988) · Zbl 0627.35076
[2] Borchers, W.; Miyakawa, T., Algebraic \(L^2\) decay for the Navier-Stokes flows in exterior domains, Acta Math., 165, 189-227 (1990) · Zbl 0722.35014
[3] Carpio, A., Large time behavior in incompressible Navier-Stokes equations, SIAM J. Math. Anal., 27, 449-475 (1996) · Zbl 0845.76019
[4] Chen, Z.-M., A sharp decay result on strong solutions of the Navier-Stokes equations in the whole space, Comm. Partial Differential Equations, 16, 801-820 (1991) · Zbl 0741.35056
[5] Chen, J.; Chen, Z.-M.; Dong, B.-Q., Existence of \(H^2\)-global attractors of two-dimensional micropolar fluid flows, J. Math. Anal. Appl., 322, 512-522 (2006) · Zbl 1098.35035
[6] Dong, B.-Q.; Chen, Z.-M., Global attractors of two-dimensional micropolar fluid flows in some unbounded domains, Appl. Math. Comput., 182, 610-620 (2006) · Zbl 1103.76008
[7] Dong, B.-Q.; Li, Y., Large time behavior to the system of incompressible non-Newtonian fluids in \(R^2\), J. Math. Anal. Appl., 298, 667-676 (2004) · Zbl 1059.35104
[8] Dong, B.-Q., Decay of solutions to equations modelling incompressible bipolar non-Newtonian fluids, Electron. J. Differential Equations, 125, 1-13 (2005) · Zbl 1091.35067
[9] Dong, B.-Q.; Chen, Z.-M., Time decay rates of non-Newtonian flows in \(R_+^n\), J. Math. Anal. Appl., 324, 820-833 (2006) · Zbl 1102.76014
[10] Eringen, A. C., Theory of micropolar fluids, J. Math. Mech., 16, 1-18 (1966) · Zbl 0145.21302
[11] Galdi, G. P.; Rionero, S., A note on the existence and uniqueness of solutions of micropolar fluid equations, Internat. J. Engrg. Sci., 14, 105-108 (1977) · Zbl 0351.76006
[12] Gallay, T.; Wayne, C., Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on \(R^2\), Arch. Ration. Mech. Anal., 163, 209-258 (2002) · Zbl 1042.37058
[13] He, C.; Miyakawa, T., Nonstationary Navier-Stokes flows in a two-dimensional exterior domain with rotational symmetries, Indiana Univ. Math. J., 55, 1483-1556 (2006) · Zbl 1122.35095
[14] Kajikiya, R.; Miyakawa, T., On \(L^2\) decay of weak solutions of Navier-Stokes equations in \(R^n\), Math. Z., 192, 135-148 (1986) · Zbl 0607.35072
[15] Kato, T., Strong \(L^p\) solutions of the Navier-Stokes equations in \(R^m\), with application to weak solutions, Math. Z., 187, 471-480 (1984) · Zbl 0545.35073
[16] Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Fluids (1969), Gordon & Breach: Gordon & Breach New York · Zbl 0184.52603
[17] Lange, H., The existence of instationary flows of incompressible micropolar fluids, Arch. Mech., 29, 741-744 (1977) · Zbl 0378.76009
[18] Leray, J., Essai sur les le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 193-248 (1934) · JFM 60.0726.05
[19] Łukaszewicz, G., Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology (1999), Birkhäuser: Birkhäuser Boston · Zbl 0923.76003
[20] Łukaszewicz, G., Long time behavior of 2D micropolar fluid flows, Math. Comput. Model., 34, 1787-1802 (2001) · Zbl 1020.76003
[21] Łukaszewicz, G.; Sadowski, W., Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phys., 55, 247-257 (2004) · Zbl 1055.35026
[22] Miyakawa, T., On upper and lower bounds of rates of decay for nonstationary Navier-Stokes flows in the whole space, Hiroshima Math. J., 32, 431-462 (2002) · Zbl 1048.35063
[23] Oliver, M.; Titi, E. S., Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in \(R^n\), J. Funct. Anal., 172, 1-18 (2000) · Zbl 0960.35081
[24] Resndiz, B.; Rojas-Medar, M., Existence of weak solution of micropolar fluid equations in a time dependent domain, Rev. Mat. Apl., 23, 27-46 (2002) · Zbl 1048.35086
[25] Sava, V. A., On the existence of solutions of the micropolar fluid equations, Rev. Roumaine Math. Pures Appl., 25, 1111-1123 (1980) · Zbl 0458.76007
[26] Schonbek, M. E., Large time behavior of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11, 733-763 (1986) · Zbl 0607.35071
[27] Schonbek, M. E., Lower bounds of rates of decay for solution to Navier-Stokes equations, J. Amer. Math. Soc., 4, 423-449 (1991) · Zbl 0739.35070
[28] Schonbek, M. E., Large time behaviour of solutions to the Navier-Stokes equations in \(H^m\) spaces, Comm. Partial Differential Equations, 20, 103-117 (1995) · Zbl 0831.35132
[29] Taylor, M., Pseudo Differential Operator and Nonlinear PDE (1991), Birkhäuser Boston: Birkhäuser Boston Cambridge · Zbl 0746.35062
[30] Teman, R., The Navier-Stokes Equations (1977), North-Holland: North-Holland Amsterdam · Zbl 0406.35053
[31] Wiegner, M., Decay results for weak solutions of the Navier-Stokes equations in \(R^n\), J. London Math. Soc., 35, 303-313 (1987) · Zbl 0652.35095
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