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A weighted L q-approach to Stokes flow around a rotating body

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Abstract

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 ω 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.

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References

  1. Bergh J. and Löfström J. (1976). Interpolation Spaces. Springer, New York

    MATH  Google Scholar 

  2. Borchers, W.: Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes-Gleichungen inkompressibler viskoser Flüssigkeiten. Habilitation Thesis, Univ. of Paderborn (1992)

  3. Brenner H. (1959). The Stokes resistance of an arbitrary particle, II. Chem. Eng. Sci. 19: 599–624

    Article  Google Scholar 

  4. Farwig R. (1992). The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces. Math. Z. 211: 409–447

    Article  MathSciNet  MATH  Google Scholar 

  5. Farwig R. (2005). An L q-analysis of viscous fluid flow past a rotating obstacle. Tôhoku Math. J. 58: 129–147

    Article  MathSciNet  Google Scholar 

  6. Farwig, R.: Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle. Banach Center Publications 70, 73–84 (Warsaw 2005)

  7. Farwig R. and Hishida T. (2007). Stationary Navier-Stokes flow around a rotating obstacle. Funkcial. Ekvac. 50: 371–403

    Article  MathSciNet  MATH  Google Scholar 

  8. Farwig R., Hishida T. and Müller D. (2004). L q-Theory of a singular “winding” integral operator arising from fluid dynamics. Pacific J. Math. 215: 297–312

    Article  MathSciNet  MATH  Google Scholar 

  9. Galdi G.P. (2002). On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications. In: Friedlander, S. and Serre, D. (eds) Handbook of Mathematical Fluid Dynamics, vol 1. Elsevier, Amsterdam

    Google Scholar 

  10. Galdi G.P. (2003). Steady flow of a Navier-Stokes fluid around a rotating obstacle. J. Elast. 71: 1–31

    Article  MathSciNet  MATH  Google Scholar 

  11. Garcia-Cuerva J. and Rubio de Francia J.L. (1985). Weighted Norm Inequalities and Related Topics. North Holland, Amsterdam

    MATH  Google Scholar 

  12. Gunther R.B., Hudspeth R.T. and Thomann E.A. (2002). Hydrodynamic flows on submerged rigid bodies–steady flow. J. Math. Fluid Mech. 4: 187–202

    Article  MathSciNet  Google Scholar 

  13. Hishida T. (2006). L q estimates of weak solutions to the stationary Stokes equations around a rotating body. J. Math. Soc. Japan 58: 743–767

    Article  MathSciNet  MATH  Google Scholar 

  14. Kirchhoff G. (1869). Über die Bewegung eines Rotationskörpers in einer Flüssigkeit. Crelle J. 71: 237–281

    Google Scholar 

  15. Kračmar S., Novotný A. and Pokorný M. (2001). Estimates of Oseen kernels in weighted L p spaces. J. Math. Soc. Japan 53: 59–111

    Article  MathSciNet  MATH  Google Scholar 

  16. Kračmar S., Nečasová Š. and Penel P. (2005). Estimates of weak solutions in anisotropically weighted Sobolev spaces to the stationary rotating Oseen equations. IASME Trans. 2: 854–861

    MathSciNet  Google Scholar 

  17. Kurtz D.S. (1980). Littlewood-Paley and multiplier theorems on weighted L p spaces. Trans. Am. Math. Soc. 259: 235–254

    Article  MathSciNet  MATH  Google Scholar 

  18. Muckenhoupt B. (1972). Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165: 207–226

    Article  MathSciNet  MATH  Google Scholar 

  19. Nečasová Š. (2005). 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

    MathSciNet  Google Scholar 

  20. Nečasová Š. (2004). Asymptotic properties of the steady fall of a body in viscous fluids. Math. Methods Appl. Sci. 27: 1969–1995

    Article  MathSciNet  MATH  Google Scholar 

  21. Rychkov V. (2001). Littlewood-Paley theory and function spaces with \({A_p^{\rm {loc}}}\) weights. Math. Nachr. 224: 145–180

    Article  MathSciNet  MATH  Google Scholar 

  22. San Martín J.A., Starovoitov V. and Tucsnak M. (2002). Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Ration. Mech. Anal. 161: 113–147

    Article  MathSciNet  MATH  Google Scholar 

  23. Sawyer E. (1986). Weighted inequalities for the one-sided Hardy-Littlewood maximal function. Trans. Am. Math. Soc. 297: 53–61

    Article  MathSciNet  MATH  Google Scholar 

  24. Serre D. (1987). Chute libre d’un solide dans un fluide visqueux incompressible. Existence. Jpn. J. Appl. Math. 4: 99–110

    MathSciNet  MATH  Google Scholar 

  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

    MATH  Google Scholar 

  27. Thomson, W. (Lord Kelvin): Mathematical and Physical Papers, vol. 4, Cambridge University Press London (1982)

  28. Weinberger H.F. (1973). On the steady fall of a body in a Navier-Stokes fluid. Proc. Symp. Pure Math. 23: 421–440

    Google Scholar 

  29. Weinberger H.F. (1972). Variational properties of steady fall in Stokes flow. J. Fluid Mech. 52: 321–344

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Darmstadt University of Technology, 64289, Darmstadt, Germany

    Reinhard Farwig

  2. Mathematical Institute of Academy of Sciences, Žitná 25, 11567, Prague 1, Czech Republic

    Miroslav Krbec & Šárka Nečasová

Authors
  1. Reinhard Farwig
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  2. Miroslav Krbec
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  3. Šárka Nečasová
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Corresponding author

Correspondence to Reinhard Farwig.

Additional information

The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan no. AV0Z10190503, by the Grant Agency of the Academy of Sciences No. IAA100190505, and by the joint research project of DAAD (D/04/25763) and the Academy of Sciences of the Czech Republic (D-CZ 3/05-06).

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Farwig, R., Krbec, M. & Nečasová, Š. A weighted L q-approach to Stokes flow around a rotating body. Ann. Univ. Ferrara 54, 61–84 (2008). https://doi.org/10.1007/s11565-008-0040-6

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  • DOI: https://doi.org/10.1007/s11565-008-0040-6

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