Skip to main content
SpringerLink
Log in

Asymptotic Behavior of a Leray Solution around a Rotating Obstacle

  • Chapter
  • First Online:
Parabolic Problems

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 80))

Abstract

We consider a body, B, that rotates, without translating, in a Navier-Stokes liquid that fills the whole space exterior to B. We analyze asymptotic properties of steady-state motions, that is, time-independent solutions to the equation of motion written in a frame attached to the body. We prove that “weak” steady-state solutions in the sense of J. Leray that satisfy the energy inequality are Physically Reasonable in the sense of R. Finn, provided the “size” of the data is suitably restricted.

Mathematics Subject Classification (2000). Primary 35Q30, 76D05; Secondary 35B40.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Wolfgang Borchers. Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes Gleichungen inkompressibler viskoser Flüssigkeiten. Habilitationsschrift, Universität Paderborn, 1992.

    Google Scholar 

  2. Reinhard Farwig. An L q-analysis of viscous fluid flow past a rotating obstacle. Tohoku Math. J. (2), 58(1):129–147, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  3. Reinhard Farwig and Toshiaki Hishida. Stationary Navier-Stokes flow around a rotating obstacle. Funkc. Ekvacioj, Ser. Int., 50(3):371–403, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  4. Reinhard Farwig, Toshiaki Hishida, and Detlef Müller. L q-theory of a singular winding integral operator arising from fluid dynamics. Pac. J. Math., 215(2):297–312, 2004.

    Article  MATH  Google Scholar 

  5. Robert Finn. On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems. Arch. Ration. Mech. Anal., 19:363–406, 1965.

    Google Scholar 

  6. Giovanni P. Galdi. On the asymptotic properties of Leray’s solutions to the exterior steady three-dimensional Navier-Stokes equations with zero velocity at infinity. Ni, Wei-Ming (ed.) et al., Degenerate diffusions. Proceedings of the IMA workshop, held at the University of Minnesota, MN, USA, from May 13 to May 18, 1991. New York: Springer-Verlag. IMA Vol. Math. Appl. 47, 95–103 (1993), 1993.

    Google Scholar 

  7. Giovanni P.Galdi. An introduction to the mathematical theory of the Navier-Stokes equations . Vol. I: Linearized steady problems. Springer Tracts in Natural Philosophy. 38. New York, NY: Springer-Verlag, 1994.

    Google Scholar 

  8. Giovanni P. Galdi. On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications. Friedlander, S. (ed.) et al., Handbook of mathematical fluid dynamics. Vol. 1. Amsterdam: Elsevier. 653–791 (2002), 2002.

    Chapter  Google Scholar 

  9. Giovanni P. Galdi and Ana L. Silvestre. The steady motion of a Navier-Stokes liquid around a rigid body. Arch. Ration. Mech. Anal., 184(3):371–400, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  10. Giovanni P. Galdi. Steady Flow of a Navier-Stokes Fluid around a Rotating Obstacle. Journal of Elasticity, 71:1–31, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  11. Toshiaki Hishida. An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle. Arch. Ration. Mech. Anal., 150(4):307–348, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  12. Toshiaki Hishida and Yoshihiro Shibata. Decay estimates of the Stokes flow around a rotating obstacle. In Kyoto Conference on the Navier-Stokes Equations and their Applications, RIMS Kôkyûroku Bessatsu, B1, pages 167–186. Res. Inst. Math. Sci. (RIMS), Kyoto, 2007.

    Google Scholar 

  13. Toshiaki Hishida and Yoshihiro Shibata. L p -L q estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle. Arch. Ration. Mech. Anal., 193(2):339–421, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  14. Stanislav Kračmar, Šárka Nečasová, and Patrick Penel. L q-approach to weak solutions of the Oseen flow around a rotating body. Rencławowicz, Joanna (ed.) et al., Parabolic and Navier-Stokes equations. Part 1. Proceedings of the confererence, Bçdlewo, Poland, September 10–17, 2006. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 81, Pt. 1, 259–276 (2008), 2008.

    Google Scholar 

  15. Jean Leray. Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl., 12:1–82, 1933.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, USA

    Giovanni P. Galdi

  2. Institut für Mathematik, RWTH-Aachen, Aachen, Germany

    Mads Kyed

Authors
  1. Giovanni P. Galdi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mads Kyed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giovanni P. Galdi .

Editor information

Editors and Affiliations

  1. Inst. Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Joachim Escher

  2. School of Physical Sciences, Dept. Mathematics, University of California, Irvine, Irvine, 92697-3875, California, USA

    Patrick Guidotti

  3. , Fachbereich Mathematik, TU Darmstadt, Schlossgartenstr 7, Darmstadt, 64289, Germany

    Matthias Hieber

  4. Inst. Applied Mathematics & Mechanics, Warsaw University, ul. Banacha 2, Warszawa, 02-097, Poland

    Piotr Mucha

  5. , FB Mathematik und Informatik, Universität Halle-Wittenberg, Theodor-Lieser-Strasse 5, Halle, 06099, Germany

    Jan W. Prüss

  6. 3-4-1, Ohkubo Shinjuku-ku, Tokyo, 169-8555, Japan

    Yoshihiro Shibata

  7. , Deparment of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA

    Gieri Simonett

  8. Inst. Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Christoph Walker

  9. Fac. Mathematics & Information Sciences, Warsaw University of Technology, pl. Politechniki 1, Warszawa, 00-661, Poland

    Wojciech Zajaczkowski

Additional information

To Professor Herbert Amann on the occasion of his 70th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Basel AG

About this chapter

Cite this chapter

Galdi, G.P., Kyed, M. (2011). Asymptotic Behavior of a Leray Solution around a Rotating Obstacle. In: Escher, J., et al. Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 80. Springer, Basel. https://doi.org/10.1007/978-3-0348-0075-4_13

Download citation

Publish with us

Policies and ethics

Search

Navigation