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Asymptotic Behavior of the Navier–Stokes Type Problem

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Fluids Under Control

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References

  1. T. K. Aldoss and T. W. Abou-Arab, Experimental study of the flow around a rotating cylinder in crossflow, Experimental Thermal and Fluid Science, 3 (1990), 316–322.

    Article  Google Scholar 

  2. A. Rosa da Silva, N. Aristeu da Silveira, M. G. de Lima and D. A. Rade, Numerical simulations of flows over a rotating circular cylinder using the immersed boundary method, J. Braz. Soc. Mech. Sci. & Eng., 33 (2011).

    Google Scholar 

  3. K. I. Babenko, On stationary solutions of the problem of flow past a body of a viscous incompressible fluid, Mat. Sb., 91 (1973), 3–26 (Russian); [English translation: Math. USSR-Sbornik, 20 (1973), 1–25.]

    Google Scholar 

  4. P., Deuring and S. Kračmar, Exterior stationary Navier-Stokes flows in 3D with non-zero velocity at infinity: Approximation by flows in bounded domains, Mathematische Nachrichten, 269/270 (2004), 86–115.

    Google Scholar 

  5. P. Deuring, S. Kračmar and Š. Nečasová, A representation formula for linearized stationary incompressible viscous flows around rotating and translating bodies, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 237–253.

    MathSciNet  MATH  Google Scholar 

  6. P. Deuring, S. Kračmar and Š. Nečasová, On pointwise decay of linearized stationary incompressible viscous flow around rotating and translating bodies, SIAM J. Math. Anal., 43 (2011), 705–738.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Deuring, S. Kračmar and Š. Nečasová, Linearized stationary incompressible flow around rotating and translating bodies: Asymptotic profile of the velocity gradient and decay estimate of the second derivatives of the velocity, J. Differential Equations, 252 (2012), 459–476.

    Article  MathSciNet  MATH  Google Scholar 

  8. P. Deuring, S. Kračmar and Š. Nečasová, A linearized system describing stationary incompressible viscous flow around rotating and translating bodies: Improved decay estimates of the velocity and its gradient, Dynamical Systems, Differential Equations and Applications, Ed. by W. Feng, Z. Feng, M. Grasselli, A. Ibragimov, X. Lu, S. Siegmund and J. Voigt. Discrete Contin. Dyn. Syst., Supplement 2011 (8th AIMS Conference, Dresden, Germany) 1 (2011), 351–361.

    Google Scholar 

  9. P. Deuring, S. Kračmar and Š. Nečasová, Pointwise decay of stationary rotational viscous incompressible flows with nonzero velocity at infinity, J. Differential Equations, 255 (2013), 1576–1606.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Deuring, S. Kračmar and Š. Nečasová, Linearized stationary incompressible flow around rotating and translating bodies—Leray solutions, Discrete Contin. Dyn. Syst., 7 (2014), 967–979.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Deuring, S. Kračmar and Š. Nečasová, Asymptotic structure of viscous incompressible flow around a rotating body, with nonvanishing flow field at infinity, Z. Angew. Math. Phys., 68, (2017), 1, 16.

    Google Scholar 

  12. P. Deuring, S. Kračmar and Š. Nečasová, A leading term for the velocity of stationary viscous incompressible flow around a rigid body performing a rotation and a translation, Discrete Contin. Dyn. Syst.,37 (2017), 3, 1389–1409.

    Google Scholar 

  13. P. Deuring, S. Kračmar and Š. Nečasová, Artificial boundary conditions for linearized stationary incompressible viscous flow around rotating and translating body, Math. Nachr., 294, (2021), 1, 56–73.

    Google Scholar 

  14. T. Eiter, G. P. Galdi, Spatial decay of the vorticity field of time-periodic viscous flow past a body, Arch. Ration. Mech. Anal., 242, (2021), 1, 149–178.

    Google Scholar 

  15. T. Eiter, M. Kyed, Viscous flow around a rigid body performing a time-periodic motion, J. Math. Fluid Mech., 23, (2021), 1, 28, 23 pp.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. R. Farwig, G. P. Galdi and M. Kyed, Asymptotic structure of a Leray solution to the Navier-Stokes flow around a rotating body, Pacific J. Math., 253 (2011), 367–382.

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Farwig, R. B. Guenther, Š. Nečasová and E. A. Thomann, The fundamental solution of the linearized instationary Navier-Stokes equations of motion around a rotating and translating body, Discrete Contin. Dyn. Syst., 34 (2014), 511–529.

    Article  MathSciNet  MATH  Google Scholar 

  20. R. Farwig and T. Hishida, Stationary Navier-Stokes flow around a rotating obstacle, Funkcialaj Ekvacioj, 50 (2007), 371–403.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Farwig and T. Hishida, Asymptotic profiles of steady Stokes and Navier-Stokes flows around a rotating obstacle, Ann. Univ. Ferrara, 55 (2009), 263–277.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. Farwig and T. Hishida, Asymptotic profile of steady Stokes flow around a rotating obstacle, Manuscripta Math., 136 (2011), 315–338.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Farwig and T. Hishida, Leading term at infinity of steady Navier-Stokes flow around a rotating obstacle, Math. Nachr., 284 (2011), 2065–2077.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. R. Farwig, An Lq-analysis of viscous fluid flow past a rotating obstacle, Tohoku Math. J., 2, 58 (2006), 1, 129–147.

    Google Scholar 

  26. R. Farwig, M. Krbec and Š. Nečasová, A weighted Lq approach to Stokes flow around a rotating body, Ann. Univ. Ferrara, Sez.,54 (2008), 61–84.

    Article  MathSciNet  MATH  Google Scholar 

  27. R. Farwig, M. Krbec and Š. Nečasová, A weighted Lq-approach to Oseen flow around a rotating body, Math. Meth. Appl. Sci., 31 (2008), 551–574.

    Article  MATH  Google Scholar 

  28. J. Feng, H. H. Hu and D. D. Joseph, Direct simulation of initial value problems for the motion of solid bodies in a Newtonian Fluid, Part 2. Couette and Poiseuille flows, Journal of Fluid Mechanics,277 (1994), 271–301.

    Google Scholar 

  29. R. Finn, Estimates at infinity for stationary solutions of the Navier-Stokes equations, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine,3 (1959), 387–418.

    MathSciNet  MATH  Google Scholar 

  30. R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Rational Mech. Anal., 19 (1965), 363–406.

    Article  MathSciNet  MATH  Google Scholar 

  31. G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems (rev. ed.), Springer, New York, 1994.

    Google Scholar 

  32. G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, Handbook of Mathematical Fluid Dynamics, Ed. by S. Friedlander, D. Serre, Elsevier, 1 (2002), 653–791.

    Google Scholar 

  33. G. P. Galdi, Steady flow of a Navier-Stokes fluid around a rotating obstacle, J. Elasticity,71 (2003), 1–31.

    Article  MathSciNet  MATH  Google Scholar 

  34. G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems (2nd ed.), Springer, New York, 2011.

    Google Scholar 

  35. G. P. Galdi and M. Kyed, Steady-state Navier-Stokes flows past a rotating body: Leary solutions are physically reasonable, Arch. Rat. Mech. Anal.,200 (2011), 21–58.

    Article  MATH  Google Scholar 

  36. G. P. Galdi and M. Kyed, Asymptotic behavior of a Leray solution around a rotating obstacle, Progress in Nonlinear Differential Equations and Their Applications,80 (2011), 251–266.

    Article  MathSciNet  MATH  Google Scholar 

  37. G. P. Galdi and M. Kyed, A simple proof of Lq-estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part I: Strong solutions, Proc. Am. Math. Soc., 141 (2013), 573–583.

    Article  MATH  Google Scholar 

  38. G. P. Galdi and M. Kyed, A simple proof of Lq-estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part II: Weak solutions, Proc. Am. Math. Soc., 141 (2013), 1313–1322.

    Article  MATH  Google Scholar 

  39. G. P. Galdi and A. L. Silvestre, The steady motion of a Navier-Stokes liquid around a rigid body, Arch. Rat. Mech. Anal., 184 (2007), 371–400.

    Article  MathSciNet  MATH  Google Scholar 

  40. G.P. Galdi, Navier-Stokes flow past a rigid body that moves by time-periodic motion, J. Math. Fluid Mech., 24 (2022), 2, 30, 23 pp.

    Google Scholar 

  41. T. Hishida, Lq estimates of weak solutions to the stationary Stokes equations around a rotating body, J. Math. Soc. Japan,58 (2006), 744–767.

    Article  MATH  Google Scholar 

  42. T. Hishida and M. Kyed, On the asymptotic structure of steady Stokes and Navier-Stokes flows around a rotating two-dimensional body, Pacific J. Math., 315 (2021), 1, 89–109.

    Google Scholar 

  43. H. Heck, H. Kim and H. Kozono, On the stationary Navier-Stokes flow around a rotating body, Manus. Math., 138 (2012), 315–345.

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Korolev and V. Šverák, On the large-distance asymptotics of steady-state solutions of the Navier-Stokes equations in 3D exterior domains, Ann. Inst. Henri Poincaré, Anal. Non Linéaire,28 (2011), 303–313.

    Article  MathSciNet  MATH  Google Scholar 

  45. S. Kračmar, M. Krbec, Š. Nečasová, P. Penel and K. Schumacher, On the Lq-approach with generalized anisotropic weights of the weak solution of the Oseen flow around a rotating body, Nonlinear Analysis,71 (2009), e2940–e2957.

    Article  MATH  Google Scholar 

  46. S. Kračmar, Š. Nečasov á and P. Penel, Estimates of weak solutions in anisotropically weighted Sobolev spaces to the stationary rotating Oseen equations, IASME Transactions,2 (2005), 854–861.

    Google Scholar 

  47. S. Kračmar, Š. Nečasov á and P. Penel, Anisotropic L2 estimates of weak solutions to the stationary Oseen type equations in \( \mathbb {R} ^{3}\) for a rotating body, RIMS Kôkyûroku Bessatsu, B1 (2007), 219–235.

    Google Scholar 

  48. S. Kračmar, Š. Nečasov á and P. Penel, Anisotropic L2 estimates of weak solutions to the stationary Oseen type equations in 3D—exterior domain for a rotating body, J. Math. Soc. Japan,62 (2010), 239–268.

    Google Scholar 

  49. S. Kračmar, A. Novotný and M. Pokorný, Estimates of Oseen kernels in weighted Lpspaces, J. Math. Soc. Japan,53 (2001), 59–111.

    Article  MathSciNet  MATH  Google Scholar 

  50. M. Kyed, On the asymptotic structure of a Navier-Stokes flow past a rotating body, J. Math. Soc. Japan,66 (2014), 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  51. M. Kyed, Periodic solutions to the Navier-Stokes equations, Habilitation thesis, Technische Universität Darmstadt, Darmstadt, 2012.

    Google Scholar 

  52. Š. Nečasová and K. Schumacher, Strong solution to the Stokes equations of a flow around a rotating body in weighted Lq spaces, Math. Nachr.,284 (2011), 1701–1714.

    Article  MathSciNet  MATH  Google Scholar 

  53. V. A. Solonnikov, A priori estimates for second order parabolic equations, Trudy Mat. Inst. Steklov., 70 (1964), 133–212 (Russian); English translation: AMS Translations, 65 (1967), 51–137.

    Google Scholar 

  54. V. A. Solonnikov, Estimates of the solutions of a nonstationary linearized system of Navier-Stokes equations, Trudy Mat. Inst. Steklov., 70 (1964), 213–317 (Russian); English translation : AMS Translations, 75 (1968), 1–116.

    Google Scholar 

  55. 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.,8 (2006), 77–98.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The research of Š. N. and S.K. were supported by the Czech Sciences Foundation (GAČR), GA19-04243S and by 22-01591S (final version). Moreover, the research of Š. N. was supported by Praemium Academiae of Š. Nečasová and RVO 67985840. S.K. was supported by CZ.02.1.01/0.0/0.0/16-019/0000778. Š. N. and S.K. would like to thank to Paul Deuring for his reading of the manuscript, point out to us various misprints and for his suggestions and support.

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, Czech Technical University, Prague, Czech Republic

    Stanislav Kračmar

  2. Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic

    Šárka Nečasová

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  1. Stanislav Kračmar
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Correspondence to Šárka Nečasová .

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Editors and Affiliations

  1. Faculty of Mechanical Engineering, Czech Technical University in Prague, Prague, Czech Republic

    Tomáš Bodnár

  2. Swanson School of Engineering, University of Pittsburgh, Pittsburgh, PA, USA

    Giovanni P. Galdi

  3. Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic

    Šárka Nečasová

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Kračmar, S., Nečasová, Š. (2023). Asymptotic Behavior of the Navier–Stokes Type Problem. In: Bodnár, T., Galdi, G.P., Nečasová, Š. (eds) Fluids Under Control. Advances in Mathematical Fluid Mechanics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-27625-5_5

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