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An \(L^q\)-analysis of viscous fluid flow past a rotating obstacle. (English) Zbl 1136.76340
Summary: Consider the problem of time-periodic strong solutions of the Stokes and Navier-Stokes system modelling viscous incompressible fluid flow past or around a rotating obstacle in Euclidean three-space. Introducing a rotating coordinate system attached to the body, a linearization yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. In this paper we find an explicit solution for the linear whole space problem when the axis of rotation is parallel to the velocity of the fluid at infinity. For the analysis of this solution in \(L^q\)-spaces, \(1<q<\infty\), we will use tools from harmonic analysis and a special maximal operator reflecting paths of fluid particles past or around the obstacle.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
35C15 Integral representations of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
76D99 Incompressible viscous fluids
76U05 General theory of rotating fluids
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[1] W. Borchers, Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes-Gleichungen inkompressibler viskoser Flüssigkeiten, Habilitation Thesis, Univ. of Paderborn, 1992.
[2] Z. M. Chen and T. Miyakawa, Decay properties of weak solutions to a perturbed Navier-Stokes system in \(\R^n\), Adv. Math. Sci. Appl. 7 (1997), 741–770. · Zbl 0893.35092
[3] R. Farwig, T. Hishida and D. Müller, \(L^q\)-theory of a singular “winding” integral operator arising from fluid dynamics, Pacific J. Math. 215 (2004), 297–312. · Zbl 1057.35028 · doi:10.2140/pjm.2004.215.297
[4] R. Farwig, The stationary Navier-Stokes equations in a \(3D\)-exterior domain, Recent topics on mathematical theory of viscous incompressible fluid (Tsukuba, 1996), 53–115, Lecture Notes in Numer. Appl. Anal. 16, Kinokuniya, Tokyo, 1998. · Zbl 0941.35064
[5] R. Farwig and H. Sohr, Weighted estimates for the Oseen equations and the Navier-Stokes equations in exterior domains, Theory of Navier-Stokes equations, 11–30, Ser. Adv. Math. Appl. Sci. 47, World Sci. Publishing, River Edge, N. J., 1998. · Zbl 0934.35120
[6] G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, Vol. I, Linearized steady problems, Springer Tracts Nat. Philos. 38, Springer-Verlag New York, 1994. · Zbl 0949.35004
[7] G. P. Galdi, On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications, Handbook of mathematical fluid dynamics, Vol. I, 653–791, North-Holland, Amsterdam, 2002. · Zbl 1230.76016
[8] G. P. Galdi, Steady flow of a Navier-Stokes fluid around a rotating obstacle, J. Elasticity 71 (2003), 1–31. · Zbl 1156.76367 · doi:10.1023/B:ELAS.0000005543.00407.5e
[9] T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Rational Mech. Anal. 150 (1999), 307–348. · Zbl 0949.35106 · doi:10.1007/s002050050190
[10] T. Hishida, The Stokes operator with rotation effect in exterior domains, Analysis (Munich) 19 (1999), 51–67. · Zbl 0938.35114 · doi:10.1524/anly.1999.19.1.51
[11] Š. Nečasova, Some remarks on the steady fall of a body in Stokes and Oseen flow, Acad. Sciences Czech Republic, Math. Institute, Preprint 143 (2001).
[12] Š. Nečasova, Asymptotic properties of the steady fall of a body in viscous fluids, Math. Methods Appl. Sci. 27 (2004), 1969–1995. · Zbl 1174.76306 · doi:10.1002/mma.467
[13] O. Sawada, The Navier-Stokes flow with linearly growing initial velocity in the whole space, Darmstadt University of Technology, Department of Mathematics, Preprint no. 2288 (2003). · Zbl 1091.35062
[14] H. Sohr, The Navier-Stokes Equations. An elementary functional analytic approach, Birkhäuser Adv. Texts, Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001.
[15] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, N. J., 1970. · Zbl 0207.13501
[16] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Ser. 43, Monographs in Harmonic Analysis III, Princeton Univ. Press, Princeton, N. J., 1993. · Zbl 0821.42001
[17] E. A. Thomann and R. B. Guenther, The fundamental solution of the linearized Navier-Stokes equations for spinning bodies in three spatial dimensions—time dependent case, J. Math. Fluid Mech., · Zbl 1125.35076 · doi:10.1007/s00021-004-0139-1
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