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A weighted $$L^q$$-approach to Oseen flow around a rotating body. (English) Zbl 1132.76015
Summary: We study time-periodic Oseen flows past a rotating body in $$\mathbb R^3$$ proving weighted a priori estimates in $$L^q$$-spaces using Muckenhoupt weights. After a time-dependant change of coordinates, the problem is reduced to a stationary Oseen equation with additional terms $$(w \wedge x) \cdot \nabla u$$ and $$-\omega \wedge u$$ in the momentum equation, where $$\omega$$ denotes the angular velocity. Due to the asymmetry of Oseen flow and to describe its wake, we use anisotropic Muckenhoupt weights, a weighted theory of Littlewood-Paley decomposition and of maximal operators as well as one-sided univariate weights, one-sided maximal operators and a new version of Jones’ factorization theorem.

##### MSC:
 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D07 Stokes and related (Oseen, etc.) flows 35Q35 PDEs in connection with fluid mechanics 76U05 General theory of rotating fluids
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##### References:
 [1] Farwig, Tôhoku Mathematics Journal 58 pp 129– (2005) [2] Estimates of Lower Order Derivatives of Viscous Fluid Flow Past a Rotating Obstacle, vol. 70. Banach Center Publications: Warsaw, 2005; 73–82. · Zbl 1101.35348 [3] , . Anisotropic L2 estimates of weak solutions to the stationary Oseen-type equations in $$\mathbb{R}$$3 for a rotating body. Preprint 165, Academy of Sciences of the Czech Republic, Mathematical Institute, 2006; Proceedings of the RIMS, Kyoto University, 2007; B1:219–235. [4] , . Lq approach of weak solutions of Oseen flow around a rotating body. Preprint, 2006, submitted. [5] . On the spectrum of an Oseen-type operator arising from flow past a rotating body. Preprint no. 2484, FB Mathematik, TU Darmstadt, 2006, submitted [6] . Stationary Navier–Stokes flow around a rotating obstacle. Preprint no. 2445, FB Mathematik, TU Darmstadt, 2006; Funkcialaj Ekvacioj, in press. [7] Farwig, Pacific Journal of Mathematics 215 pp 297– (2004) [8] , . A weighted Lq approach to Stokes flow around a rotating body. Preprint no. 2422, FB Mathematik, TU Darmstadt, 2005, submitted. [9] Hishida, Journal of the Mathematical Society of Japan 58 pp 743– (2006) [10] . Lp–Lq estimate of the Stokes operator and Navier–Stokes flows in the exterior of a rotating obstacle. Preprint, 2006. [11] Some remarks on the steady fall of a body in Stokes and Oseen flow. Preprint 143, Academy of Sciences of the Czech Republic, Mathematical Institute, 2001; Proceedings of AIMS Conference, 2006. [12] Nečasova, IASME Transactions 2 pp 1262– (2005) [13] Nečasova, Mathematical Methods in the Applied Sciences 27 pp 1969– (2004) [14] On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications. In Handbook of Mathematical Fluid Dynamics, (eds), vol. 1. Elsevier: Amsterdam, 2002. [15] Kračmar, IASME Transactions 2 pp 854– (2005) [16] Farwig, Mathematische Zeitschrift 211 pp 409– (1992) [17] . Weighted Norm Inequalities and Related Topics. North-Holland: Amsterdam, 1985. [18] Muckenhoupt, Transactions of the American Mathematical Society 165 pp 207– (1972) [19] Kurtz, Transactions of the American Mathematical Society 259 pp 235– (1980) [20] Sawyer, Transactions of the American Mathematical Society 297 pp 53– (1986) [21] Rychkov, Mathematische Nachrichten 224 pp 145– (2001) [22] . Weighted Hardy Spaces. Lecture Notes in Mathematics, vol. 1381. Springer: Berlin, 1989. · Zbl 0676.42021 · doi:10.1007/BFb0091154 [23] Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press: Princeton, NJ, 1993. [24] . Interpolation Spaces. Springer: New York, 1976. · doi:10.1007/978-3-642-66451-9 [25] Kračmar, Journal of the Mathematical Society of Japan 53 pp 59– (2001)
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