×

zbMATH — the first resource for mathematics

Exponential decay of the vorticity in the steady-state flow of a viscous liquid past a rotating body. (English) Zbl 1342.35244
Summary: Consider the flow of a Navier-Stokes liquid past a body rotating with a prescribed constant angular velocity, \(\omega\), and assume that the motion is steady with respect to a body-fixed frame. In this paper we show that the vorticity field associated to every “weak” solution corresponding to data of arbitrary “size” (Leray Solution) must decay exponentially fast outside the wake region at sufficiently large distances from the body. Our result improves and generalizes in a non-trivial way famous results by D. C. Clark [Indiana Univ. Math. J. 20, 633–654 (1971; Zbl 0187.24506)] and K. I. Babenko and M. M. Vasil’ev [J. Appl. Math. Mech. 37, 651–665 (1973); translation from Prikl. Mat. Mekh. 37, 690–705 (1973; Zbl 0295.76015)] obtained in the case \(\omega=0\).
MSC:
35Q35 PDEs in connection with fluid mechanics
76U05 General theory of rotating fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Amrouche, C.; Consiglieri, L., On the stationary Oseen equations in \({\mathbb{R}^3}\), Commun. Math. Anal., 10, 5-29, (2011) · 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. 91(133), 3-26 (1973, Russian) [English translation: Math. USSR-Sbornik20, 1-25 (1973)] · Zbl 1125.35076
[3] Babenko, K.I., Vasil’ev, M.M.: On the asymptotic behavior of a steady flow of viscous fluid at some distance from an immersed body. Prikl. Mat. Meh. 37, 690-705 (1973, Russian) [English translation: J. Appl. Math. Mech. 37, 651-665 (1973)] · Zbl 1296.35122
[4] Batchelor G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (2002) · Zbl 0958.76001
[5] Borchers, W.: Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes Gleichungen inkompressibler viskoser Flüssigkeiten. Habilitationsschrift, University of Paderborn, 1992 · Zbl 1273.35214
[6] Clark, D., The vorticity at infinity for solutions of the stationary Navier-Stokes equations in exterior domains, Indiana Univ. Math. J., 20, 633-654, (1971) · Zbl 0187.24506
[7] 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
[8] 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
[9] 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. Differ. Equ., 252, 459-476, (2012) · Zbl 1238.35097
[10] 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 and its gradient, Discrete Contin. Dyn. Syst. Suppl., 2011, 351-361, (2011) · Zbl 1306.35083
[11] Deuring, P.; Kračmar, S.; Necasová, Š., Pointwise decay of stationary rotational viscous incompressible flows with nonzero velocity at infinity, J. Differ. Equ., 255, 1576-1606, (2013) · Zbl 1284.35306
[12] Deuring, P.; Kračmar, S.; Nečasová, Š., Linearized stationary incompressible flow around rotating and translating bodies—leray solutions, Discrete Contin. Dyn. Syst. Ser. S, 7, 967-979, (2014) · Zbl 1304.35536
[13] Deuring, P., Kračmar, S., Nečasová, Š.: Asymptotic structure of viscous incompressible flow around a rotating body, with nonvanishing flow field at infinity (submitted) · Zbl 1367.35105
[14] 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
[15] Farwig, R., An \({L^{q}}\)-analysis of viscous fluid flow past a rotating obstacle, Tôhoku Math. J., 58, 129-147, (2006) · Zbl 1136.76340
[16] Farwig, R.: Estimates of Lower Order Derivatives of Viscous Fluid Flow Past a Rotating Obstacle, Vol. 70. Banach Center Publications, pp. 73-84, 2005 · Zbl 1101.35348
[17] Farwig, R.; Galdi, G.P.; Kyed, M., Asymptotic structure of a Leray solution to the Navier-Stokes flow around a rotating body, Pac. J. Math., 253, 367-382, (2011) · Zbl 1234.35035
[18] Farwig, R.; Hishida, T., Stationary Navier-Stokes flow around a rotating obstacle, Funkcialaj Ekvacioj, 50, 371-403, (2007) · Zbl 1180.35408
[19] 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
[20] Farwig, R.; Hishida, T., Asymptotic profile of steady Stokes flow around a rotating obstacle, Manuscr. Math., 136, 315-338, (2011) · Zbl 1229.35172
[21] 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
[22] Farwig, R.; Hishida, T.; Müller, D., \({L^q}\)-theory of a singular “winding” integral operator arising from fluid dynamics, Pac. J. Math., 215, 297-312, (2004) · Zbl 1057.35028
[23] 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
[24] 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
[25] Farwig, R.; Neustupa, J., On the spectrum of a Stokes-type operator arising from flow around a rotating body, Manuscr. Math., 122, 419-437, (2007) · Zbl 1126.35050
[26] Finn, R., On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Rational Mech. Anal., 19, 363-406, (1965) · Zbl 0149.44606
[27] Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. II. Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, Vol. 39. Springer, New York, 1994 · Zbl 0949.35004
[28] Galdi, G.P.: On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications. Handbook of Mathematical Fluid Dynamics, Vol. I (Eds. Friedlander S. and Serre D.). North-Holland, Amsterdam, pp. 653-791, 2002 · Zbl 1230.76016
[29] Galdi, G.P., Steady flow of a Navier-Stokes fluid around a rotating obstacle, J. Elast., 71, 1-31, (2003) · Zbl 1156.76367
[30] Galdi, G.P.: Further properties of steady-state solutions to the Navier-Stokes problem past a three-dimensional obstacle. J. Math. Phys. 48 (2007) · Zbl 1144.81345
[31] Galdi G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2nd edn. Springer, New York (2011) · Zbl 1245.35002
[32] Galdi, G.P.: Further properties of weak solutions to the steady-state Navier-Stokes problem around a rotating body, lecture held at the Workshop “Navier-Stokes Equations”, RWTH Aachen, 2013 · Zbl 1132.76015
[33] Galdi, G.P.; Heywood, J.G.; Shibata, Y., On the global existence and convergence to steady state of Navier-Stokes flow past an obstacle that is started from the rest, Arch. Rational Mech. Anal., 138, 307-319, (1997) · Zbl 0898.35071
[34] Galdi, G.P.; Kyed, M., Steady-state Navier-Stokes flows past a rotating body: Leray solutions are physically reasonable, Arch. Rational Mech. Anal., 200, 21-58, (2011) · Zbl 1229.35176
[35] Galdi, G.P.; Kyed, M., Asymptotic behavior of a Leray solution around a rotating obstacle, Progress Nonlinear Differ. Equ. Appl., 60, 251-266, (2011) · Zbl 1247.35168
[36] 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. Am. Math. Soc., 141, 573-583, (2013) · Zbl 1261.35106
[37] 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. Am. Math. Soc., 141, 1313-1322, (2013) · Zbl 1260.35111
[38] Galdi, G.P.; Silvestre, A.L., Strong solutions to the Navier-Stokes equations around a rotating obstacle, Arch. Rational Mech. Anal., 176, 331-350, (2005) · Zbl 1081.35076
[39] Galdi, G.P.; Silvestre, A.L., The steady motion of a Navier-Stokes liquid around a rigid body, Arch. Rational Mech. Anal., 184, 371-400, (2007) · Zbl 1111.76010
[40] Galdi, G.P.; Silvestre, A.L., 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) · Zbl 1119.76011
[41] 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
[42] Hishida, T., An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Rational Mech. Anal., 150, 307-348, (1999) · Zbl 0949.35106
[43] Hishida, T., The Stokes operator with rotating effect in exterior domains, Analysis, 19, 51-67, (1999) · Zbl 0938.35114
[44] 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
[45] Hishida, T.; Shibata, Y., Decay estimates of the Stokes flow around a rotating obstacle, RIMS Kôkyûroku Bessatsu, B1, 167-186, (2007) · Zbl 1119.35052
[46] Hishida, T.; Shibata, Y., \({L_p}\) -\({L_q}\) estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle, Arch. Rational Mech. Anal., 193, 339-421, (2009) · Zbl 1169.76015
[47] 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)
[48] 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
[49] 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
[50] 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
[51] Kračmar, S.; Penel, P., Variational properties of a generic model equation in exterior 3D domains, Funkcialaj Ekvacioj, 47, 499-523, (2004) · Zbl 1114.35053
[52] Kračmar, S.; Penel, P., New regularity results for a generic model equation in exterior 3D domains, Banach Center Publ. Warsaw, 70, 139-155, (2005) · Zbl 1101.35350
[53] Kyed, M., Asymptotic profile of a linearized flow past a rotating body, Q. Appl. Math., 71, 489-500, (2013) · Zbl 1273.35214
[54] Kyed, M., On a mapping property of the Oseen operator with rotation, Discrete Contin. Dyn. Syst. Ser. S, 6, 1315-1322, (2013) · Zbl 1260.35117
[55] Kyed, M., On the asymptotic structure of a Navier-Stokes flow past a rotating body, J. Math. Soc. Japan, 66, 1-16, (2014) · Zbl 1296.35122
[56] Ladyzhenskaya, O.A.: Investigation of the Navier-Stokes equation for stationary motion of an incompressible fluid. Uspekhi Mat. Nauk14(87), 75-97 (1959, Russian) · Zbl 0100.09602
[57] Leray, J., Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’ hydrodynamique, J. Math. Pures Appl., 12, 1-82, (1933) · Zbl 0006.16702
[58] 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
[59] 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)
[60] 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
[61] Solonnikov, V.A.: A priori estimates for second order parabolic equations. Trudy Mat. Inst. Steklov., 70, 133-212 (1964, Russian) [English translation: AMS Transl. 65, 51-137 (1967)] · Zbl 0168.08202
[62] Thomann, E.A.; Guenther, R.B., 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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.