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A weighted \(L^q\)-approach to Stokes flow around a rotating body. (English) Zbl 1248.35158
Summary: Considering time-periodic Stokes flow around a rotating body in \({\mathbb R^2}\) or \({\mathbb R^3}\) we prove weighted a priori estimates in \(L^q\)-spaces for the whole space problem. After a time-dependent change of coordinates the problem is reduced to a stationary Stokes equation with the additional term \({(\omega \times x)\cdot\nabla u}\) in the equation of momentum, where \(\omega \) denotes the angular velocity. In cylindrical coordinates attached to the rotating body we allow for Muckenhoupt weights which may be anisotropic or even depend on the angular variable and prove weighted \(L^q\)-estimates using the weighted theory of Littlewood-Paley decomposition and of maximal operators.

35Q35 PDEs in connection with fluid mechanics
35B45 A priori estimates in context of PDEs
46N20 Applications of functional analysis to differential and integral equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
Full Text: DOI
[1] Bergh J. and Löfström J. (1976). Interpolation Spaces. Springer, New York · Zbl 0344.46071
[2] Borchers, W.: Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes-Gleichungen inkompressibler viskoser Flüssigkeiten. Habilitation Thesis, Univ. of Paderborn (1992) · Zbl 0655.76022
[3] Brenner, H., The Stokes resistance of an arbitrary particle, II, Chem. Eng. Sci., 19, 599-624, (1959)
[4] 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
[5] Farwig, R., An \(L\)\^{\(q\)}-analysis of viscous fluid flow past a rotating obstacle, Tôhoku Math. J., 58, 129-147, (2005) · Zbl 1136.76340
[6] Farwig, R.: Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle. Banach Center Publications 70, 73-84 (Warsaw 2005) · Zbl 1101.35348
[7] Farwig, R.; Hishida, T., Stationary Navier-Stokes flow around a rotating obstacle, Funkcial. Ekvac., 50, 371-403, (2007) · Zbl 1180.35408
[8] 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
[9] Galdi, G.P.; Friedlander, S. (ed.); Serre, D. (ed.), On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications, (2002), Amsterdam
[10] Galdi, G.P., Steady flow of a Navier-Stokes fluid around a rotating obstacle, J. Elast., 71, 1-31, (2003) · Zbl 1156.76367
[11] Garcia-Cuerva J. and Rubio de Francia J.L. (1985). Weighted Norm Inequalities and Related Topics. North Holland, Amsterdam · Zbl 0578.46046
[12] Gunther, R.B.; Hudspeth, R.T.; Thomann, E.A., Hydrodynamic flows on submerged rigid bodies-steady flow, J. Math. Fluid Mech., 4, 187-202, (2002) · Zbl 1011.35103
[13] Hishida, T., \(L\)\^{\(q\)} estimates of weak solutions to the stationary Stokes equations around a rotating body, J. Math. Soc. Japan, 58, 743-767, (2006) · Zbl 1184.35241
[14] Kirchhoff, G., Über die bewegung eines rotationskörpers in einer flüssigkeit, Crelle J., 71, 237-281, (1869) · JFM 02.0731.01
[15] 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
[16] 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)
[17] Kurtz, D.S., Littlewood-Paley and multiplier theorems on weighted \(L\)\^{\(p\)} spaces, Trans. Am. Math. Soc., 259, 235-254, (1980) · Zbl 0436.42012
[18] Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function, Trans. Am. Math. Soc., 165, 207-226, (1972) · Zbl 0236.26016
[19] 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)
[20] 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
[21] Rychkov, V., Littlewood-Paley theory and function spaces with \({A_p^{\rm{loc}}}\) weights, Math. Nachr., 224, 145-180, (2001) · Zbl 0984.42011
[22] San Martín, J.A.; Starovoitov, V.; Tucsnak, M., Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161, 113-147, (2002) · Zbl 1018.76012
[23] Sawyer, E., Weighted inequalities for the one-sided Hardy-Littlewood maximal function, Trans. Am. Math. Soc., 297, 53-61, (1986) · Zbl 0627.42009
[24] Serre, D., Chute libre d’un solide dans un fluide visqueux incompressible, Existence. Jpn. J. Appl. Math., 4, 99-110, (1987) · Zbl 0655.76022
[25] Strömberg, J.O., Torchinsky, A.: Weighted Hardy Spaces. Lecture Notes in Math., vol. 1381, Springer, Berlin (1989)
[26] Stein E.M. (1993). Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton · Zbl 0821.42001
[27] Thomson, W. (Lord Kelvin): Mathematical and Physical Papers, vol. 4, Cambridge University Press London (1982) · Zbl 1136.76340
[28] Weinberger, H.F., On the steady fall of a body in a Navier-Stokes fluid, Proc. Symp. Pure Math., 23, 421-440, (1973)
[29] Weinberger, H.F., Variational properties of steady fall in Stokes flow, J. Fluid Mech., 52, 321-344, (1972) · Zbl 0245.76023
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