Abstract
We consider weak (“Leray”) solutions to the stationary Navier–Stokes system with Oseen and rotational terms, in an exterior domain. It is shown the velocity may be split into a constant times the first column of the fundamental solution of the Oseen system, plus a remainder term decaying pointwise near infinity at a rate which is higher than the decay rate of the Oseen tensor. This result improves the theory by Kyed (Q Appl Math 71:489–500, 2013).
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Deuring, P.: The single-layer potential associated with the time-dependent Oseen system. In: Proceedings of the 2006 IASME/WSEAS International Conference on Continuum Mechanics. Chalkida, Greeece, May 11–13, 2006, pp. 117–125 (2006)
Deuring, P., Kračmar, S.: Exterior stationary Navier-Stokes flows in 3D with non-zero velocity at infinity: approximation by flows in bounded domains. Math. Nachr. 269–270, 86–115 (2004)
Deuring, P., Kračmar, S., Nečasová, Š.: A representation formula for linearized stationary incompressible viscous flows around rotating and translating bodies. Discrete Contin. Dyn. Syst. Ser. S 3, 237–253 (2010)
Deuring, P., Kračmar, S., Nečasová, Š.: On pointwise decay of linearized stationary incompressible viscous flow around rotating and translating bodies. SIAM J. Math. Anal. 43, 705–738 (2011)
Deuring, P., Kračmar, S., Nečasová, Š.: 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. Differ. Equ. 252, 459–476 (2012)
Deuring, P., Kračmar, S., Nečasová, Š.: A linearized system describing stationary incompressible viscous flow around rotating and translating bodies: improved decay estimates of the velocity and its gradient. Discrete Contin. Dyn. Syst. 2011, 351–361 (2011)
Deuring, P., Kračmar, S., Nečasová, Š.: Pointwise decay of stationary rotational viscous incompressible flows with nonzero velocity at infinity. J. Differ. Equ. 255, 1576–1606 (2013)
Deuring, P., Kračmar, S., Nečasová, Š.: Linearized stationary incompressible flow around rotating and translating bodies-Leray solutions. Discrete Contin. Dyn. Syst. Ser. S 7, 967–979 (2014)
Deuring, P., Kračmar, Š., Necasová, Š.: A leading term for the velocity of stationary viscous incompressible flow performing a rotation and a translation. Discrete Contin. Dyn. Syst. Ser. A 37, 261–281 (2017)
Deuring, P., Varnhorn, W.: On Oseen resolvent estimates. Differ. Integral Equ. 23, 1139–1149 (2010)
Eidelman, S.D.: On fundamental solutions of parabolic systems II. Mat. Zb. 53, 73–136 (1961). (Russian)
Farwig, R.: The stationary exterior 3D-problem of Oseen and Navier–Stokes equations in anisotropically weighted Sobolev spaces. Math. Z. 211, 409–447 (1992)
Farwig, R.: An \(L^{q}\)-analysis of viscous fluid flow past a rotating obstacle. Tôhoku Math. J. 58, 129–147 (2006)
Farwig, R.: Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle. Banach Cent. Publ. 70, 73–84 (2005)
Farwig, R., Galdi, G.P., Kyed, M.: Asymptotic structure of a Leray solution to the Navier–Stokes flow around a rotating body. Pacific J. Math. 253, 367–382 (2011)
Farwig, R., Hishida, T.: Stationary Navier–Stokes flow around a rotating obstacle. Funkc. Ekvacioj 50, 371–403 (2007)
Farwig, R., Hishida, T.: Asymptotic profiles of steady Stokes and Navier–Stokes flows around a rotating obstacle. Ann. Univ. Ferrara Sez. VII 55, 263–277 (2009)
Farwig, R., Hishida, T.: Asymptotic profile of steady Stokes flow around a rotating obstacle. Manuscr. Math. 136, 315–338 (2011)
Farwig, R., Hishida, T.: Leading term at infinity of steady Navier–Stokes flow around a rotating obstacle. Math. Nachr. 284, 2065–2077 (2011)
Farwig, R., Hishida, T., Müller, D.: \(L^q\)-theory of a singular “winding” integral operator arising from fluid dynamics. Pac. J. Math. 215, 297–312 (2004)
Farwig, R., Krbec, M., Nečasová, Š.: A weighted \(L^q\) approach to Stokes flow around a rotating body. Ann. Univ. Ferrara Sez. VII. 54, 61–84 (2008)
Farwig, R., Krbec, M., Nečasová, Š.: A weighted \(L^q\)-approach to Oseen flow around a rotating body. Math. Methods Appl. Sci. 31, 551–574 (2008)
Farwig, R., Neustupa, J.: On the spectrum of a Stokes-type operator arising from flow around a rotating body. Manuscr. Math. 122, 419–437 (2007)
Farwig, R., Neustupa, J.: On the spectrum of an Oseen-type operator arising from fluid flow past a rotating body in \(L^q_{\sigma }(\Omega )\). Tohoku Math. J. 62, 287–309 (2010)
Galdi, G.P.: An introduction to the mathematical theory of the Navier–Stokes equations. I. Linearised steady problems, Springer tracts in natural philosophy, vol. 38. Springer, New York (1998)
Galdi, G.P.: On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics, pp. 653–791. North-Holland, Amsterdam (2002)
Galdi, G.P.: Steady flow of a Navier–Stokes fluid around a rotating obstacle. J. Elast. 71, 1–31 (2003)
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, 2nd edn. Springer, New York (2011)
Galdi, G.P., Kyed, M.: Steady-state Navier–Stokes flows past a rotating body: Leray solutions are physically reasonable. Arch. Ration. Mech. Anal. 200, 21–58 (2011)
Galdi, G.P., Kyed, M.: Asymptotic behavior of a Leray solution around a rotating obstacle. Prog. Nonlinear Diff. Equ. Appl. 60, 251–266 (2011)
Galdi, G.P., Kyed, M.: A simple proof of \(L^q\)-estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part I: strong solutions. Proc. Am. Math. Soc. 141, 573–583 (2013)
Galdi, G.P., Kyed, M.: A simple proof of \(L^q\)-estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part II: weak solutions. Proc. Am. Math. Soc. 141, 1313–1322 (2013)
Galdi, G.P., Silvestre, A.L.: Strong solutions to the Navier–Stokes equations around a rotating obstacle. Arch. Ration. Mech. Anal. 176, 331–350 (2005)
Galdi, G.P., Silvestre, A.L.: The steady motion of a Navier–Stokes liquid around a rigid body. Arch. Ration. Mech. Anal. 184, 371–400 (2007)
Galdi, G.P., Silvestre, A.L.: Further results on steady-state flow of a Navier–Stokes liquid around a rigid body. Existence of the wake. RIMS Kôkyûroku Bessatsu B1, 108–127 (2008)
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)
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)
Hishida, T.: The Stokes operator with rotating effect in exterior domains. Analysis 19, 51–67 (1999)
Hishida, T.: \(L^q\) estimates of weak solutions to the stationary Stokes equations around a rotating body. J. Math. Soc. Jpn. 58, 744–767 (2006)
Hishida, T., Shibata, Y.: Decay estimates of the Stokes flow around a rotating obstacle. RIMS Kôkyûroku Bessatsu B1, 167–186 (2007)
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)
Kračmar, S., Krbec, M., Nečasová, Š., Penel, P., Schumacher, K.: On the \(L^q\)-approach with generalized anisotropic weights of the weak solution of the Oseen flow around a rotating body. Nonlinear Anal. 71, e2940–e2957 (2009)
Kračmar, S., Nečasová, Š., Penel, P.: Estimates of weak solutions in anisotropically weighted Sobolev spaces to the stationary rotating Oseen equations. IASME Trans. 2, 854–861 (2005)
Kračmar, S., Nečasová, Š., Penel, P.: Anisotropic \(L^2\) estimates of weak solutions to the stationary Oseen type equations in \( \mathbb{R} ^{3}\) for a rotating body. RIMS Kôkyûroku Bessatsu B1, 219–235 (2007)
Kračmar, S., Nečasová, Š., Penel, P.: Anisotropic \(L^2\) estimates of weak solutions to the stationary Oseen type equations in 3D-exterior domain for a rotating body. J. Math. Soc. Jpn. 62, 239–268 (2010)
Kračmar, S., Novotný, A., Pokorný, M.: Estimates of Oseen kernels in weighted \(L^p\) spaces. J. Math. Soc. Jpn. 53, 59–111 (2001)
Kračmar, S., Penel, P.: Variational properties of a generic model equation in exterior 3D domains. Funkc. Ekvacioj 47, 499–523 (2004)
Kračmar, S., Penel, P.: New regularity results for a generic model equation in exterior 3D domains. Banach Center Publ. Wars. 70, 139–155 (2005)
Kyed, M.: Periodic Solutions to the Navier–Stokes Equations, Habilitation Thesis. Technische Universität Darmstadt, Darmstadt (2012)
Kyed, M.: Asymptotic profile of a linearized flow past a rotating body. Q. Appl. Math. 71, 489–500 (2013)
Kyed, M.: On a mapping property of the Oseen operator with rotation. Discrete Contin. Dyn. Syst. Ser. S 6, 1315–1322 (2013)
Kyed, M.: On the asymptotic structure of a Navier–Stokes flow past a rotating body. J. Math. Soc. Jpn. 66, 1–16 (2014)
Neri, U.: Singular Integrals. Lecture Notes in Mathematics, vol. 200. Springer, Berlin (1971)
Nečasová, Š.: Asymptotic properties of the steady fall of a body in viscous fluids. Math. Methods Appl. Sci. 27, 1969–1995 (2004)
Nečasová, Š.: 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 (2005)
Nečasová, Š., Schumacher, K.: Strong solution to the Stokes equations of a flow around a rotating body in weighted \(L^q\) spaces. Math. Nachr. 284, 1701–1714 (2011)
Solonnikov, V.A.: A priori estimates for second order parabolic equations, Trudy Mat. Inst. Steklov. 70:133–212 (1964) (Russian); English translation: AMS Translations 65:51–137 (1967)
Solonnikov, V.A.: Estimates of the solutions of a nonstationary linearized system of Navier–Stokes equations, Trudy Mat. Inst. Steklov. 70:213–317 (1964) (Russian); English translation: AMS Translations 75:1–116 (1968)
Stein, E.M.: Singular Integrals and Differentiability of Functions. Princeton University Press, Princeton (1970)
Thomann, E.A., Guenther, R.B.: The fundamental solution of the linearized Navier–Stokes equations for spinning bodies in three spatial dimensions - time dependent case. J. Math. Fluid Mech. 8, 77–98 (2006)
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Deuring, P., Kračmar, S. & Nečasová, Š. Asymptotic structure of viscous incompressible flow around a rotating body, with nonvanishing flow field at infinity. Z. Angew. Math. Phys. 68, 16 (2017). https://doi.org/10.1007/s00033-016-0760-x
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DOI: https://doi.org/10.1007/s00033-016-0760-x