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Decay Estimates of Gradient of a Generalized Oseen Evolution Operator Arising from Time-Dependent Rigid Motions in Exterior Domains

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Abstract

Let us consider the motion of a viscous incompressible fluid past a rotating rigid body in three dimensions, where the translational and angular velocities of the body are prescribed but time-dependent. In a reference frame attached to the body, we have the Navier–Stokes system with the drift and (one half of the) Coriolis terms in a fixed exterior domain. The existence of the evolution operator T(t, s) in the space \(L^q\) generated by the linearized non-autonomous system was proved by Hansel and Rhandi (J Reine Angew Math 694:1–26, 2014) and the large time behavior of T(t, s)f in \(L^r\) for \((t-s)\rightarrow \infty \) was then developed by Hishida (Math Ann 372:915–949, 2018) when f is taken from \(L^q\) with \(q\leqq r\). The contribution of the present paper concerns such \(L^q\)-\(L^r\) decay estimates of \(\nabla T(t,s)\) with optimal rates, which must be useful for the study of stability/attainability of the Navier–Stokes flow in several physically relevant situations. Our main theorem completely recovers the \(L^q\)-\(L^r\) estimates for the autonomous case (Stokes and Oseen semigroups, those semigroups with rotating effect) in three dimensional exterior domains, which were established by Hishida and Shibata (Arch Ration Mech Anal 193:339–421, 2009), Iwashita (Math Ann 285, 265–288, 1989), Kobayashi and Shibata (Math Ann 310:1–45, 1998), Maremonti and Solonnikov (Ann Sc Norm Super Pisa 24:395–449, 1997) and Shibata (in: Amann, Arendt, Hieber, Neubrander, Nicaise, von Below (eds) Functional analysis and evolution equations, the Günter Lumer volume. Birkhäuser, Basel, pp 595–611, 2008).

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References

  1. Bergh, J., Löfström, J.: Interpolation Spaces. Springer, Berlin 1976

    MATH  Google Scholar 

  2. Bogovskiĭ, M.E.: Solution of the first boundary value problem for the equation of continuity of an incompressible medium. Soviet Math. Dokl. 20, 1094–1098, 1979

    Google Scholar 

  3. Borchers, W., Sohr, H.: On the equations \(\text{ rot } v=g\) and \(\text{ div } u=f\) with zero boundary conditions. Hokkaido Math. J. 19, 67–87, 1990

    MathSciNet  MATH  Google Scholar 

  4. Chen, Z.M., Miyakawa, T.: Decay properties of weak solutions to a perturbed Navier–Stokes system in \({\mathbb{R}}^n\). Adv. Math. Sci. Appl. 7, 741–770, 1997

    MathSciNet  MATH  Google Scholar 

  5. Dan, W., Shibata, Y.: On the \(L_q\)-\(L_r\) estimates of the Stokes semigroup in a two-dimensional exterior domain. J. Math. Soc. Jpn. 51, 181–207, 1999

    MATH  Google Scholar 

  6. Dan, W., Shibata, Y.: Remark on the \(L_q\)-\(L_\infty \) estimate of the Stokes semigroup in a \(2\)-dimensional exterior domain. Pac. J. Math. 189, 223–239, 1999

    MATH  Google Scholar 

  7. Enomoto, Y., Shibata, Y.: Local energy decay of solutions to the Oseen equation in the exterior domains. Indiana Univ. Math. J. 53, 1291–1330, 2004

    MathSciNet  MATH  Google Scholar 

  8. Enomoto, Y., Shibata, Y.: On the rate of decay of the Oseen semigroup in exterior domains and its applications to the Navier–Stokes equation. J. Math. Fluid Mech. 7, 339–367, 2005

    ADS  MathSciNet  MATH  Google Scholar 

  9. Farwig, R., Galdi, G.P., Kyed, M.: Asymptotic structure of a Leray solution to the Navier–Stokes flow around a rotating body. Pac. J. Math. 253, 367–382, 2011

    MathSciNet  MATH  Google Scholar 

  10. Farwig, R., Hishida, T.: Leading term at infinity of steady Navier–Stokes flow around a rotating obstacle. Math. Nachr. 284, 2065–2077, 2011

    MathSciNet  MATH  Google Scholar 

  11. Farwig, R., Neustupa, J.: Spectral properties in \(L^q\) of an Oseen operator modelling fluid flow past a rotating body. Tohoku Math. J. 62, 287–309, 2010

    MathSciNet  MATH  Google Scholar 

  12. Farwig, R., Sohr, H.: Weighted \(L^q\)-theory for the Stokes resolvent in exterior domains. J. Math. Soc. Jpn. 49, 251–288, 1997

    MATH  Google Scholar 

  13. Finn, R.: Stationary solutions of the Navier–Stokes equations. Proc. Symp. Appl. Math. 17, 121–153, 1965

    MATH  Google Scholar 

  14. Fujiwara, D.: \(L^p\)-theory for characterizing the domain of the fractional powers of \(-\Delta \) in the half space. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 15, 169–177, 1968

    MATH  Google Scholar 

  15. Fujiwara, D., Morimoto, H.: An \(L_r\)-theorem of the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24, 685–700, 1977

    MathSciNet  MATH  Google Scholar 

  16. Galdi, G.P.: On the Motion of a Rigid Body in a Viscous Liquid: A Mathematical Analysis with Applications. Handbook of Mathematical Fluid Dynamics, vol. I, pp. 653–791. North-Holland, Amsterdam 2002

    MATH  Google Scholar 

  17. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Steady-State Problems, Second edn. Springer, Berlin 2011

    MATH  Google Scholar 

  18. Galdi, G.P.: Viscous flow past a body translating by time-periodic motion with zero average. Arch. Ration. Mech. Anal. (to appear)

  19. Galdi, G.P., Heywood, J.G., Shibata, Y.: On the global existence and convergence to steady state of Navier–Stokes flow past an obstacle that is started from rest. Arch. Ration. Mech. Anal. 138, 307–318, 1997

    MathSciNet  MATH  Google Scholar 

  20. Galdi, G.P., Hishida, T.: Attainability of Time-Periodic Flow of a Viscous Liquid Past an Oscillating Body. arXiv:2001.07292.

  21. Geissert, M., Hansel, T.: A non-autonomous model problem for the Oseen–Navier–Stokes flow with rotating effect. J. Math. Soc. Jpn. 63, 1027–1037, 2011

    MathSciNet  MATH  Google Scholar 

  22. Geissert, M., Heck, H., Hieber, M.: \(L^p\)-theory of the Navier–Stokes flow in the exterior of a moving or rotating obstacle. J. Reine Angew. Math. 596, 45–62, 2006

    MathSciNet  MATH  Google Scholar 

  23. Geissert, M., Heck, H., Hieber, M.: On the equation div \(u=g\) and Bogovskiĭ’s operator in Sobolev spaces of negative order. Oper. Theory Adv. Appl. 168, 113–121, 2006

    MATH  Google Scholar 

  24. Giga, Y.: Domains of fractional powers of the Stokes operator in \(L_r\) spaces. Arch. Ration. Mech. Anal. 89, 251–265, 1985

    MATH  Google Scholar 

  25. Hansel, T.: On the Navier–Stokes equations with rotating effect and prescribed outflow velocity. J. Math. Fluid Mech. 13, 405–419, 2011

    ADS  MathSciNet  MATH  Google Scholar 

  26. Hansel, T., Rhandi, A.: Non-autonomous Ornstein–Uhlenbeck equations in exterior domains. Adv. Differ. Equ. 16, 201–220, 2011

    MathSciNet  MATH  Google Scholar 

  27. Hansel, T., Rhandi, A.: The Oseen–Navier–Stokes flow in the exterior of a rotating obstacle: the non-autonomous case. J. Reine Angew. Math. 694, 1–26, 2014

    MathSciNet  MATH  Google Scholar 

  28. Hishida, T.: An existence theorem for the Navier–Stokes flow in the exterior of a rotating obstacle. Arch. Ration. Mech. Anal. 150, 307–348, 1999

    MathSciNet  MATH  Google Scholar 

  29. Hishida, T.: The nonstationary Stokes and Navier–Stokes flows through an aperture. Contributions to Current Challenges in Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics (Eds. Galdi G.P., Heywood J.G. and Rannacher, R.) Birkhäuser, Basel, 79–123, 2004

  30. Hishida, T.: On the Relation Between the Large Time Behavior of the Stokes Semigroup and the Decay of Steady Stokes Flow at Infinity, Parabolic Problems: The Herbert Amann Festschrift, Progress in Nonlinear Differential Equations and Their Applications, vol. 80, pp. 343–355. Springer, Berlin 2011

    MATH  Google Scholar 

  31. Hishida, T.: Mathematical analysis of the equations for incompressible viscous fluid around a rotating obstacle. Sugaku Expos. 26, 149–179, 2013

    MATH  Google Scholar 

  32. Hishida, T.: \(L^q\)-\(L^r\) estimate of the Oseen flow in plane exterior domains. J. Math. Soc. Jpn. 68, 295–346, 2016

    MATH  Google Scholar 

  33. Hishida, T.: Stationary Navier–Stokes flow in exterior domains and Landau solutions. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (Eds. Giga Y. and Novotny A.) Springer, Berlin, 299–339, 2018

  34. Hishida, T.: Large time behavior of a generalized Oseen evolution operator, with applications to the Navier–Stokes flow past a rotating obstacle. Math. Ann. 372, 915–949, 2018

    MathSciNet  MATH  Google Scholar 

  35. Hishida, T.: \(L^q\)-\(L^r\) estimate of a generalized Oseen evolution operator, with applications to the Navier–Stokes flow past a rotating obstacle. RIMS Kokyuroku2107, 139–152, 2019

    Google Scholar 

  36. Hishida, T., Maremonti, P.: Navier–Stokes flow past a rigid body: attainability of steady solutions as limits of unsteady weak solutions, starting and landing cases. J. Math. Fluid Mech. 20, 771–800, 2018

    ADS  MathSciNet  MATH  Google Scholar 

  37. Hishida, T., Shibata, Y.: \(L_p\)-\(L_q\) estimate of the Stokes operator and Navier–Stokes flows in the exterior of a rotating obstacle. Arch. Ration. Mech. Anal. 193, 339–421, 2009

    MathSciNet  MATH  Google Scholar 

  38. Iwashita, H.: \(L_q\)-\(L_r\) estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier–Stokes initial value problems in \(L_q\) spaces. Math. Ann. 285, 265–288, 1989

    MathSciNet  MATH  Google Scholar 

  39. Kato, T.: Strong \(L^p\) solutions of the Navier–Stokes equation in \({\mathbb{R}}^m\), with applications to weak solutions. Math. Z. 187, 471–480, 1984

    MathSciNet  MATH  Google Scholar 

  40. Kobayashi, T., Shibata, Y.: On the Oseen equation in the three dimensional exterior domains. Math. Ann. 310, 1–45, 1998

    MathSciNet  MATH  Google Scholar 

  41. Korolev, A., Šverak, V.: On the large-distance asymptotics of steady state solutions of the Navier–Stokes equations in 3D exterior domains. Ann. Inst. Henri Poincare, Anal. Non Lineaire28, 303–313, 2011

    ADS  MathSciNet  MATH  Google Scholar 

  42. Maekawa, Y.: On local energy decay estimate of the Oseen semigroup in two dimensions and its application. J. Inst. Math. Jussieu 2019. https://doi.org/10.1017/s1474748019000355

    Article  Google Scholar 

  43. Maremonti, P., Solonnikov, V.A.: On nonstationary Stokes problems in exterior domains. Ann. Sc. Norm. Super. Pisa24, 395–449, 1997

    MathSciNet  MATH  Google Scholar 

  44. Miyakawa, T.: On nonstationary solutions of the Navier–Stokes equations in an exterior domain. Hiroshima Math. J. 12, 115–140, 1982

    MathSciNet  MATH  Google Scholar 

  45. Shibata, Y.: On the Oseen semigroup with rotating effect. Functional Analysis and Evolution Equations, The Günter Lumer Volume (Eds. Amann H., Arendt W., Hieber M., Neubrander F., Nicaise S. and von Below J.) Birkhäuser, Basel, 595–611, 2008

  46. Shibata, Y.: On a \(C^0\)Semigroup Associated with a Modified Oseen Equation with Rotating Effect. Advances in Mathematical Fluid Mechanics, pp. 513–551. Springer, Berlin 2010

  47. Shibata, Y.: On the \(L_p\)-\(L_q\) decay estimate for the Stokes equations with free boundary conditions in an exterior domain. Asymptot. Anal. 107, 33–72, 2018

    MathSciNet  MATH  Google Scholar 

  48. Simader, C.G., Sohr, H.: A new approach to the Helmholtz decomposition and the Neumann problem in \(L^q\)-spaces for bounded and exterior domains. Mathematical Problems Relating to the Navier–Stokes Equations. Ser. Adv. Math. Appl. Sci., Vol. 11. World Scientific Publication, River Edge, 1992

  49. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton 1993

    MATH  Google Scholar 

  50. Takahashi, T.: Attainability of a stationary Navier–Stokes flow around a rigid body rotating from rest. arXiv:2004.00781.

  51. Tanabe, H.: Equations of Evolution. Pitman, London 1979

    MATH  Google Scholar 

  52. Torchinsky, A.: Real-Variable Methods in Harmonic Analysis. Academic Press, London 1986

    MATH  Google Scholar 

  53. Yamazaki, M.: The Navier–Stokes equations in the weak-\(L^n\) space with time-dependent external force. Math. Ann. 317, 635–675, 2000

    MathSciNet  MATH  Google Scholar 

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  1. Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan

    Toshiaki Hishida

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Correspondence to Toshiaki Hishida.

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Communicated by V. Šverák

Dedicated to Professor Yoshio Yamada on his 70th birthday

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Partially supported by the Grant-in-aid for Scientific Research 18K03363 from JSPS.

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Hishida, T. Decay Estimates of Gradient of a Generalized Oseen Evolution Operator Arising from Time-Dependent Rigid Motions in Exterior Domains. Arch Rational Mech Anal 238, 215–254 (2020). https://doi.org/10.1007/s00205-020-01541-3

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