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Pointwise decay of stationary rotational viscous incompressible flows with nonzero velocity at infinity. (English) Zbl 1284.35306
Summary: We consider a stationary viscous incompressible flow around a translating and rotating body. Optimal rates of decay are derived for the velocity and its gradient, on the basis of a representation formula involving a fundamental solution constructed by E. A. Thomann and R. B. Guenther [J. Math. Fluid Mech. 8, No. 1, 77–98 (2006; Zbl 1125.35076)], for a linearized system.

MSC:
35Q30 Navier-Stokes equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
35A08 Fundamental solutions to PDEs
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[1] Amrouche, C.; Consiglieri, L., On the stationary Oseen equations in \(\mathbb{R}^3\), Commun. Math. Anal., 10, 5-29, (2010) · Zbl 1235.35216
[2] Babenko, K. I., On stationary solutions of the problem of flow past a body of a viscous incompressible fluid, Mat. Sb., Math. USSR Sb., 20, 1-25, (1973), (in Russian); English translation: · Zbl 0285.76009
[3] Deuring, P., Spatial decay of time-dependent incompressible Navier-Stokes flows with nonzero velocity at infinity, SIAM J. Math. Anal., (2013), in press · Zbl 1294.35059
[4] Deuring, P., Pointwise spatial decay of time-dependent Oseen flows: the case of data with noncompact support, Discrete Contin. Dyn. Syst. Ser. A, 33, 2757-2776, (2013) · Zbl 1295.35359
[5] Deuring, P., The single-layer potential associated with the time-dependent Oseen system, (Proceedings of the 2006 IASME/WSEAS International Conference on Continuum Mechanics, (2006), Chalkida Greece), 117-125
[6] Deuring, P.; Kračmar, S.; Nečasová, Š., A representation formula for linearized stationary incompressible viscous flows around rotating and translating bodies, Discrete Contin. Dyn. Syst. Ser. S, 3, 237-253, (2010) · Zbl 1193.35127
[7] Deuring, P.; Kračmar, S.; Nečasová, Š., On pointwise decay of linearized stationary incompressible viscous flow around rotating and translating bodies, SIAM J. Math. Anal., 43, 705-738, (2011) · Zbl 1231.35143
[8] Deuring, P.; Kračmar, S.; Nečasová, Š., Linearized stationary incompressible flow around rotating and translating bodies: asymptotic profile of the velocity gradient and decay estimate of the second derivatives of the velocity, J. Differential Equations, 252, 459-476, (2012) · Zbl 1238.35097
[9] Deuring, P.; Kračmar, S.; Nečasová, Š., A linearized system describing stationary incompressible viscous flow around rotating and translating bodies: improved decay estimates of the velocity, Discrete Contin. Dyn. Syst., Supplement, 351-361, (2011) · Zbl 1306.35083
[10] P. Deuring, S. Kračmar, Š. Nečasová, Linearized stationary incompressible flow around rotating and translating bodies - Leray solution, submitted for publication.
[11] Farwig, R., The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., 211, 409-447, (1992) · Zbl 0727.35106
[12] Farwig, R., An \(L^q\)-analysis of viscous fluid flow past a rotating obstacle, Tôhoku Math. J., 58, 129-147, (2005) · Zbl 1136.76340
[13] Farwig, R., Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle, Banach Center Publ., vol. 70, 73-84, (2005) · Zbl 1101.35348
[14] Farwig, R.; Galdi, G. P.; Kyed, M., Asymptotic structure of a Leray solution to the Navier-Stokes flow around a rotating body, Pacific J. Math., 253, 367-382, (2011) · Zbl 1234.35035
[15] R. Farwig, R.B. Guenther, Š. Nečasová, E.A. Thomann, The fundamental solution of the linearized instationary Navier-Stokes equations of motion around a rotating and translating body, submitted for publication. · Zbl 1280.35087
[16] Farwig, R.; Hishida, T., Stationary Navier-Stokes flow around a rotating obstacle, Funkcial. Ekvac., 50, 371-403, (2007) · Zbl 1180.35408
[17] Farwig, R.; Hishida, T., Asymptotic profiles of steady Stokes and Navier-Stokes flows around a rotating obstacle, Ann. Univ. Ferrara Sez. VII, 55, 263-277, (2009) · Zbl 1205.35191
[18] Farwig, R.; Hishida, T., Asymptotic profile of steady Stokes flow around a rotating obstacle, Manuscripta Math., 136, 315-338, (2011) · Zbl 1229.35172
[19] Farwig, R.; Hishida, T., Leading term at infinity of steady Navier-Stokes flow around a rotating obstacle, Math. Nachr., 284, 2065-2077, (2011) · Zbl 1229.35173
[20] Farwig, R.; Hishida, T.; Müller, D., \(L^q\)-theory of a singular “winding“ integral operator arising from fluid dynamics, Pacific J. Math., 215, 297-312, (2004) · Zbl 1057.35028
[21] Farwig, R.; Krbec, M.; Nečasová, Š., A weighted \(L^q\) approach to Stokes flow around a rotating body, Ann. Univ. Ferrara Sez. VII, 54, 61-84, (2008) · Zbl 1248.35158
[22] Farwig, R.; Krbec, M.; Nečasová, Š., A weighted \(L^q\)-approach to Oseen flow around a rotating body, Math. Methods Appl. Sci., 31, 551-574, (2008) · Zbl 1132.76015
[23] Farwig, R.; Neustupa, J., On the spectrum of a Stokes-type operator arising from flow around a rotating body, Manuscripta Math., 122, 419-437, (2007) · Zbl 1126.35050
[24] Galdi, G. P., An introduction to the mathematical theory of the Navier-Stokes equations, vol. I. linearized steady problems, (1998), Springer New York
[25] Galdi, G. P., An introduction to the mathematical theory of the Navier-Stokes equations, vol. II. nonlinear steady problems, (1994), Springer New York · Zbl 0949.35005
[26] Galdi, G. P., On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, (Friedlander, S.; Serre, D., Handbook of Mathematical Fluid Dynamics, vol. 1, (2002), Elsevier) · Zbl 1230.76016
[27] Galdi, G. P., Steady flow of a Navier-Stokes fluid around a rotating obstacle, J. Elasticity, 71, 1-31, (2003) · Zbl 1156.76367
[28] Galdi, G. P., An introduction to the mathematical theory of the Navier-Stokes equations. steady-state problems, (2011), Springer New York · Zbl 1245.35002
[29] Galdi, G. P.; Kyed, M., Steady-state Navier-Stokes flows past a rotating body: Leray solutions are physically reasonable, Arch. Ration. Mech. Anal., 200, 21-58, (2011) · Zbl 1229.35176
[30] Galdi, G. P.; Kyed, M., Asymptotic behavior of a Leray solution around a rotating obstacle, Progr. Nonlinear Differential Equations Appl., vol. 60, 251-266, (2011) · Zbl 1247.35168
[31] Galdi, G. P.; Kyed, M., A simple proof of \(L^q\)-estimates for the steady-state Oseen and Stokes equations in a rotating frame. part I: strong solutions, Proc. Amer. Math. Soc., (2013), in press · Zbl 1261.35106
[32] Galdi, G. P.; Kyed, M., A simple proof of \(L^q\)-estimates for the steady-state Oseen and Stokes equations in a rotating frame. part II: weak solutions, Proc. Amer. Math. Soc., (2013), in press · Zbl 1260.35111
[33] Galdi, G. P.; Silvestre, A. S., Strong solutions to the Navier-Stokes equations around a rotating obstacle, Arch. Ration. Mech. Anal., 176, 331-350, (2005) · Zbl 1081.35076
[34] Galdi, G. P.; Silvestre, S. A., The steady motion of a Navier-Stokes liquid around a rigid body, Arch. Ration. Mech. Anal., 184, 371-400, (2007) · Zbl 1111.76010
[35] Galdi, G. P.; Silvestre, S. A., Further results on steady-state flow of a Navier-Stokes liquid around a rigid body. existence of the wake, RIMS Kôkyûroku Bessatsu, B1, 108-127, (2008)
[36] Geissert, M.; Heck, H.; Hieber, M., \(L^p\) theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle, J. Reine Angew. Math., 596, 45-62, (2006) · Zbl 1102.76015
[37] Guenther, R. B.; Thomann, E. A., The fundamental solution of the linearized Navier-Stokes equations for spinning bodies in three spatial dimensions - time dependent case, J. Math. Fluid Mech., 8, 77-98, (2006) · Zbl 1125.35076
[38] Hishida, T., An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Ration. Mech. Anal., 150, 307-348, (1999) · Zbl 0949.35106
[39] Hishida, T., The Stokes operator with rotating effect in exterior domains, Analysis, 19, 51-67, (1999) · Zbl 0938.35114
[40] Hishida, T., \(L^q\) estimates of weak solutions to the stationary Stokes equations around a rotating body, J. Math. Soc. Japan, 58, 744-767, (2006) · Zbl 1184.35241
[41] Hishida, T.; Shibata, Y., \(L_p\)-\(L_q\) estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle, RIMS Kôkyûroku Bessatsu, B1, 167-188, (2007)
[42] Kračmar, S.; Krbec, M.; Nečasová, Š.; Penel, P.; Schumacher, K., On the \(L^q\)-approach with generalized anisotropic weights of the weak solution of the Oseen flow around a rotating body, Nonlinear Anal., 71, e2940-e2957, (2009) · Zbl 1239.76020
[43] Kračmar, S.; Nečasová, Š.; Penel, P., Estimates of weak solutions in anisotropically weighted Sobolev spaces to the stationary rotating Oseen equations, IASME Trans., 2, 854-861, (2005)
[44] Kračmar, S.; Nečasová, Š.; Penel, P., Anisotropic \(L^2\) estimates of weak solutions to the stationary Oseen type equations in \(\mathbb{R}^3\) for a rotating body, RIMS Kôkyûroku Bessatsu, B1, 219-235, (2007) · Zbl 1153.35060
[45] Kračmar, S.; Nečasová, Š.; Penel, P., Anisotropic \(L^2\) estimates of weak solutions to the stationary Oseen type equations in 3D - exterior domain for a rotating body, J. Math. Soc. Japan, 62, 239-268, (2010) · Zbl 1186.35163
[46] Kračmar, S.; Novotný, A.; Pokorný, M., Estimates of Oseen kernels in weighted \(L^p\) spaces, J. Math. Soc. Japan, 53, 59-111, (2001) · Zbl 0988.76021
[47] Kračmar, S.; Penel, P., Variational properties of a generic model equation in exterior 3D domains, Funkcial. Ekvac., 47, 499-523, (2004) · Zbl 1114.35053
[48] Kračmar, S.; Penel, P., New regularity results for a generic model equation in exterior 3D domains, Banach Center Publ., vol. 70, 139-155, (2005) · Zbl 1101.35350
[49] Kyed, M., Asymptotic profile of a linearized flow past a rotating body, Quart. Appl. Math., (2013), in press · Zbl 1273.35214
[50] Kyed, M., On the asymptotic structure of a Navier-Stokes flow past a rotating body, J. Math. Soc. Japan, (2013), in press
[51] Kyed, M., On a mapping property of the Oseen operator with rotation, Discrete Contin. Dyn. Syst. Ser. S, (2013), in press · Zbl 1260.35117
[52] Lizorkin, P. I., On multipliers of Fourier integrals in the spaces \(L_{p, \vartheta}\), Tr. Mat. Inst. Steklova, Proc. Steklov Inst. Math., 89, 269-290, (1968), (in Russian); English translation: · Zbl 0167.12404
[53] Nečasová, Š., Asymptotic properties of the steady fall of a body in viscous fluids, Math. Methods Appl. Sci., 27, 1969-1995, (2004) · Zbl 1174.76306
[54] Nečasová, Š., On the problem of the Stokes flow and Oseen flow in \(\mathbb{R}^3\) with Coriolis force arising from fluid dynamics, IASME Trans., 2, 1262-1270, (2005)
[55] Nečasová, Š; Schumacher, K., Strong solution to the Stokes equations of a flow around a rotating body in weighted \(L^q\) spaces, Math. Nachr., 284, 1701-1714, (2011) · Zbl 1291.76081
[56] Solonnikov, V. A., A priori estimates for second order parabolic equations, Tr. Mat. Inst. Steklova, Amer. Math. Soc. Transl., 65, 51-137, (1967), (in Russian); English translation: · Zbl 0179.42901
[57] Stein, E. M., Singular integrals and differentiability of functions, (1970), Princeton University Press Princeton, NJ · Zbl 0207.13501
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