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The Navier-Stokes Flow in the Exterior of Rotating Obstacles

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Nonlinear Elliptic and Parabolic Problems

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 64))

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Abstract

In this note we describe recent results on the equations of Navier-Stokes in the exterior of a rotating domain. After rewriting the problem on a fixed exterior domain Ω in ℝn, it is shown that for initial data u0 ∈ Lσp (Ω) with p ≥ n and which are satisfying a certain compatibility condition there exists a unique local mild solution to the Navier-Stokes problem. In the case of the whole space of ℝn, this local mild solution is even analytic in the space variable x.

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References

  1. H. Amann, On the strong solvability of the Navier-Stokes equations. J. Math. Fluid Mech., 2 (2000), 16–98.

    Article  MATH  MathSciNet  Google Scholar 

  2. W. Arendt, Ch. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel, 2001.

    Google Scholar 

  3. A. Banin, A. Mahalov, and B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains. Indiana Univ. Math. J. 48 (1999), 1133–1176.

    MathSciNet  Google Scholar 

  4. A. Banin, A. Mahalov and B. Nicolaenko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity. Indiana Univ. Math. J. 50 (2001), 1–35.

    MathSciNet  Google Scholar 

  5. M.E. Bogovskiı˘, Solution of the first boundary value problem for an equation of continuity of an incompressible medium. Dokl. Akad. Nauk SSSR 248 (1979), 1037–1040

    MathSciNet  Google Scholar 

  6. W. Borchers, Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes-Gleichungen inkompressibler viskoser Flüssigkeiten, Habilitationschrift Universität Paderborn, 1992.

    Google Scholar 

  7. W. Borchers, and H. Sohr,On the semigroup of the Stokes operator for exterior domains in Lq-spaces. Math. Z. 196, (1987), 415–425.

    Article  MathSciNet  Google Scholar 

  8. M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations. Handbook of Mathematical Fluid Dynamics, vol 3 (eds. S. Friedlander, D. Serre), Elsevier, 2003.

    Google Scholar 

  9. Z. Chen, and T. Miyakawa, Decay properties of weak solutions to a perturbed Navier-Stokes system in ℝn. Adv. Math. Sci. Appl. 7 (1997), 741–770.

    MathSciNet  Google Scholar 

  10. G. Da Prato, and A. Lunardi, On the Ornstein-Uhlenbeck operator in spaces of continuous functions. J. Func. Anal. 131 (1995), 94–114.

    Google Scholar 

  11. R. Farwig, T. Hishida, and D. Müller, Lq-theory of a singular ‘winding’ integral operator arising from fluid dynamics, Preprint, TU Darmstadt, 2003.

    Google Scholar 

  12. G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, Vol. 38, Springer 1998.

    Google Scholar 

  13. G.P. Galdi, Steady flow of a Navier-Stokes fluid around a rotating obstacle, J. Elasticity 71 (2003), no. 1–3, 1–31.

    MATH  MathSciNet  Google Scholar 

  14. M. Geissert, H. Heck, and M. Hieber, Lp-Theory of the Navier-Stokes flow past rotating or moving obstacles, Preprint (2004).

    Google Scholar 

  15. M. Geissert, H. Heck, and M. Hieber, Remarks on Bogovskii’s operator in Sobolev spaces of negative order, Preprint (2004).

    Google Scholar 

  16. Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L r spaces, Math. Z. 178 (1981), no. 3, 297–329.

    Article  MATH  MathSciNet  Google Scholar 

  17. Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system. J. Differential Equations, 62 (1986), 186–212.

    Article  MathSciNet  Google Scholar 

  18. Y. Giga, S. Matsui and O. Sawada, Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity. J. Math. Fluid Mech., 3 (2001), 302–315.

    MathSciNet  Google Scholar 

  19. Y. Giga, and T. Miyakawa, Solutions in Lr of the Navier-Stokes initial value problem. Arch. Rational Mech. Anal., 89 (1985), 267–281.

    MathSciNet  Google Scholar 

  20. Y. Giga, H. Sohr, On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo, 36, (1989), 313–333.

    MathSciNet  Google Scholar 

  21. Y. Giga and O. Sawada, On regularizing-decay rate estimate for solutions to the Navier-Stokes initial value problem. Nonlinear Anal. and Appl., (2003), to appear.

    Google Scholar 

  22. M. Hieber, and O. Sawada, The Navier-Stokes equations in ℝn with linearly growing initial data. Arch. Rational Mech. Anal., to appear 2004.

    Google Scholar 

  23. T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle. Arch. Rat. Mech. Anal., 150 (1999), 307–348.

    Article  MATH  MathSciNet  Google Scholar 

  24. T. Hishida, The stokes operator with rotation effect in exterior domains. Analysis, 19 (1999), 51–67.

    MATH  MathSciNet  Google Scholar 

  25. T. Kato, Strong Lp-solutions of Navier-Stokes equations in ℝn with applications to weak solutions. Math. Z., 187 (1984), 471–480.

    Article  MATH  MathSciNet  Google Scholar 

  26. O. Layzhenskaya, The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, 1969.

    Google Scholar 

  27. A. Lunardi, On the Ornstein-Uhlenbeck operator in L2-spaces with respect to invariant measures. Trans. Amer. Math. Soc. 349 (1997), 155–169.

    Article  MATH  MathSciNet  Google Scholar 

  28. A. Majda, Vorticity and the mathematical theory of incompressible fluid flow. Comm. Pure Appl. Math., 34 (1986), 187–220.

    MathSciNet  Google Scholar 

  29. G. Metafune, J. Prüss, A. Rhandi, and R. Schnaubelt, The domain of the Ornstein-Uhlenbeck operator on an Lp-space with invariant measure. Ann. Sc. Norm. Super. Pisa Cl. Sci., 1 (2002), 471–485.

    MathSciNet  Google Scholar 

  30. H. Sohr, The Navier-Stokes equations, Birkhäuser Advanced Texts: Birkhäuser, Basel, 2001.

    Google Scholar 

  31. V.A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations. J. Soviet Math., 8 (1977), 467–529.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, D-64289, Darmstadt, Germany

    Matthias Hieber

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  1. Matthias Hieber
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Editor information

Editors and Affiliations

  1. Universitá Pierre et Marie Curie, Paris

    Haim Brezis

  2. Rutgers University, New Brunswick, N.J.

    Haim Brezis

  3. Angewandte Mathematik, Universität Zurich, Winterthurerstr. 190, 8057, Zürich, Switzerland

    Michel Chipot

  4. Institute of Applied Mathematics, University of Hannover, Welfengarten 1, 30167, Hannover, Germany

    Joachim Escher

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© 2005 Birkhäuser Verlag Basel/Switzerland

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Hieber, M. (2005). The Navier-Stokes Flow in the Exterior of Rotating Obstacles. In: Brezis, H., Chipot, M., Escher, J. (eds) Nonlinear Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 64. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7385-7_13

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