# American Institute of Mathematical Sciences

2014, 7(5): 967-979. doi: 10.3934/dcdss.2014.7.967

## Linearized stationary incompressible flow around rotating and translating bodies -- Leray solutions

 1 Univ Lille Nord de France, 59000 Lille 2 Department of Technical Mathematics, Czech Technical University, Karlovo nám. 13, 121 35 Prague 2 3 Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1

Received  April 2013 Published  May 2014

We consider Leray solutions of the Oseen system with rotational terms, in an exterior domain. Such solutions are characterized by square-integrability of the gradient of the velocity and local square-integrability of the pressure. In a previous paper, we had shown a pointwise decay result for a slightly stronger type of solution. Here this result is extended to Leray solutions. We thus present a second access to this result, besides the one in G. P. Galdi, M. Kyed, Arch. Rat. Mech. Anal., 200 (2011), 21-58.
Citation: Paul Deuring, Stanislav Kračmar, Šárka Nečasová. Linearized stationary incompressible flow around rotating and translating bodies -- Leray solutions. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 967-979. doi: 10.3934/dcdss.2014.7.967
##### References:
 [1] 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. doi: 10.3934/dcdss.2010.3.237. [2] 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. doi: 10.1137/100786198. [3] 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. doi: 10.1016/j.jde.2011.08.037. [4] 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,, in Discrete Contin. Dyn. Syst. 2011, (2011), 351. [5] P. Deuring, S. Kračmar and Š. Nečasová, Pointwise decay of stationary rotational viscous incompressible flows with nonzero velocity at infinity,, J. of Differential Equations, 255 (2013), 1576. doi: 10.1016/j.jde.2013.05.016. [6] R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces,, Math. Z., 211 (1992), 409. doi: 10.1007/BF02571437. [7] R. Farwig, An $L^q$-analysis of viscous fluid flow past a rotating obstacle,, Tôhoku Math. J., 58 (2006), 1. doi: 10.2748/tmj/1145390210. [8] R. Farwig, Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle,, Banach Center Publications, 70 (2005), 73. doi: 10.4064/bc70-0-5. [9] 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. doi: 10.2140/pjm.2011.253.367. [10] R. Farwig and T. Hishida, Stationary Navier-Stokes flow around a rotating obstacle,, Funkcialaj Ekvacioj, 50 (2007), 371. doi: 10.1619/fesi.50.371. [11] R. Farwig and T. Hishida, Asymptotic profiles of steady Stokes and Navier-Stokes flows around a rotating obstacle,, Ann. Univ. Ferrara, 55 (2009), 263. doi: 10.1007/s11565-009-0072-6. [12] R. Farwig and T. Hishida, Asymptotic profile of steady Stokes flow around a rotating obstacle,, Manuscripta Math., 136 (2011), 315. doi: 10.1007/s00229-011-0479-0. [13] R. Farwig and T. Hishida, Leading term at infinity of steady Navier-Stokes flow around a rotating obstacle,, Math. Nachr., 284 (2011), 2065. doi: 10.1002/mana.200910192. [14] R. Farwig, T. Hishida and D. Müller, $L^q$-theory of a singular "winding'' integral operator arising from fluid dynamics,, Pacific J. Math., 215 (2004), 297. doi: 10.2140/pjm.2004.215.297. [15] R. Farwig, M. Krbec and Š. Nečasová, A weighted $L^q$ approach to Stokes flow around a rotating body,, Ann. Univ. Ferrara, 54 (2008), 61. doi: 10.1007/s11565-008-0040-6. [16] R. Farwig, M. Krbec and Š. Nečasová, A weighted $L^q$-approach to Oseen flow around a rotating body,, Math. Meth. Appl. Sci., 31 (2008), 551. doi: 10.1002/mma.925. [17] R. Farwig and J. Neustupa, On the spectrum of a Stokes-type operator arising from flow around a rotating body,, Manuscripta Math., 122 (2007), 419. doi: 10.1007/s00229-007-0078-2. [18] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearised Steady Problems,, Springer Tracts in Natural Philosophy, (1998). doi: 10.1007/978-1-4612-5364-8. [19] G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications,, in Handbook of Mathematical Fluid Dynamics. Vol. I (eds. S. Friedlander and D. Serre), (2002), 653. [20] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems,, 2nd edition, (2011). doi: 10.1007/978-0-387-09620-9. [21] G. P. Galdi and M. Kyed, Steady-state Navier-Stokes flows past a rotating body: Leray solutions are physically reasonable,, Arch. Rat. Mech. Anal., 200 (2011), 21. doi: 10.1007/s00205-010-0350-6. [22] G. P. Galdi and M. Kyed, Asymptotic behavior of a Leray solution around a rotating obstacle,, Progress in Nonlinear Differential Equations and Their Applications, 60 (2011), 251. doi: 10.1007/978-3-0348-0075-4_13. [23] G. P. Galdi and M. Kyed, 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 (2013), 573. doi: 10.1090/S0002-9939-2012-11638-7. [24] G. P. Galdi and M. Kyed, 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 (2013), 1313. doi: 10.1090/S0002-9939-2012-11640-5. [25] G. P. Galdi and A. S. Silvestre, Strong solutions to the Navier-Stokes equations around a rotating obstacle,, Arch. Rat. Mech. Anal., 176 (2005), 331. doi: 10.1007/s00205-004-0348-z. [26] G. P. Galdi and S. A. Silvestre, The steady motion of a Navier-Stokes liquid around a rigid body,, Arch. Rat. Mech. Anal., 184 (2007), 371. doi: 10.1007/s00205-006-0026-4. [27] G. P. Galdi and S. A. Silvestre, Further results on steady-state flow of a Navier-Stokes liquid around a rigid body. Existence of the wake,, RIMS Kôkyûroku Bessatsu, B1 (2008), 108. [28] R. B. Guenther and E. A. Thomann, 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. doi: 10.1007/s00021-004-0139-1. [29] T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle,, Arch. Rat. Mech. Anal., 150 (1999), 307. doi: 10.1007/s002050050190. [30] T. Hishida, The Stokes operator with rotating effect in exterior domains,, Analysis, 19 (1999), 51. doi: 10.1524/anly.1999.19.1.51. [31] T. Hishida, $L^q$ estimates of weak solutions to the stationary Stokes equations around a rotating body,, J. Math. Soc. Japan, 58 (2006), 744. doi: 10.2969/jmsj/1156342036. [32] T. Hishida and Y. Shibata, $L_p$-$L_q$ estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle,, RIMS Kôkyûroku Bessatsu, B1 (2007), 167. [33] S. Kračmar, M. Krbec, Š. Nečasová, P. Penel and K. Schumacher, On the $L^q$-approach with generalized anisotropic weights of the weak solution of the Oseen flow around a rotating body,, Nonlinear Analysis, 71 (2009). doi: 10.1016/j.na.2009.07.017. [34] S. Kračmar, Š. Nečasová and P. Penel, Estimates of weak solutions in anisotropically weighted Sobolev spaces to the stationary rotating Oseen equations,, IASME Transactions, 2 (2005), 854. [35] S. Kračmar, Š. Nečasová and P. Penel, Anisotropic $L^2$ estimates of weak solutions to the stationary Oseen type equations in $\mathbbR ^{3}$ for a rotating body,, RIMS Kôkyûroku Bessatsu, B1 (2007), 219. [36] S. Kračmar, Š. Nečasová and P. Penel, Anisotropic $L^2$ 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. doi: 10.2969/jmsj/06210239. [37] S. Kračmar and P. Penel, Variational properties of a generic model equation in exterior 3D domains,, Funkcialaj Ekvacioj, 47 (2004), 499. doi: 10.1619/fesi.47.499. [38] S. Kračmar and P. Penel, New regularity results for a generic model equation in exterior 3D domains,, Banach Center Publications Warsaw, 70 (2005), 139. doi: 10.4064/bc70-0-9. [39] M. Kyed, Asymptotic profile of a linearized flow past a rotating body,, Quart. Appl. Math., 71 (2013), 489. doi: 10.1090/S0033-569X-2013-01288-7. [40] M. Kyed, On the asymptotic structure of a Navier-Stokes flow past a rotating body,, to appear in J. Math. Soc. Japan., (). doi: 10.2969/jmsj/06610001. [41] M. Kyed, On a mapping property of the Oseen operator with rotation,, Discrete Contin. Dynam. Syst. - Ser. S., 6 (2013), 1315. doi: 10.3934/dcdss.2013.6.1315. [42] Š. Nečasová, Asymptotic properties of the steady fall of a body in viscous fluids,, Math. Meth. Appl. Sci., 27 (2004), 1969. doi: 10.1002/mma.467. [43] Š. Nečasová, On the problem of the Stokes flow and Oseen flow in $\mathbbR^{3}$ with Coriolis force arising from fluid dynamics,, IASME Transaction, 2 (2005), 1262. [44] Š. Nečasová, K. Schumacher, Strong solution to the Stokes equations of a flow around a rotating body in weighted $L^q$ spaces,, Math. Nachr., 284 (2011), 1701. doi: 10.1002/mana.200810166.

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##### References:
 [1] 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. doi: 10.3934/dcdss.2010.3.237. [2] 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. doi: 10.1137/100786198. [3] 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. doi: 10.1016/j.jde.2011.08.037. [4] 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,, in Discrete Contin. Dyn. Syst. 2011, (2011), 351. [5] P. Deuring, S. Kračmar and Š. Nečasová, Pointwise decay of stationary rotational viscous incompressible flows with nonzero velocity at infinity,, J. of Differential Equations, 255 (2013), 1576. doi: 10.1016/j.jde.2013.05.016. [6] R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces,, Math. Z., 211 (1992), 409. doi: 10.1007/BF02571437. [7] R. Farwig, An $L^q$-analysis of viscous fluid flow past a rotating obstacle,, Tôhoku Math. J., 58 (2006), 1. doi: 10.2748/tmj/1145390210. [8] R. Farwig, Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle,, Banach Center Publications, 70 (2005), 73. doi: 10.4064/bc70-0-5. [9] 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. doi: 10.2140/pjm.2011.253.367. [10] R. Farwig and T. Hishida, Stationary Navier-Stokes flow around a rotating obstacle,, Funkcialaj Ekvacioj, 50 (2007), 371. doi: 10.1619/fesi.50.371. [11] R. Farwig and T. Hishida, Asymptotic profiles of steady Stokes and Navier-Stokes flows around a rotating obstacle,, Ann. Univ. Ferrara, 55 (2009), 263. doi: 10.1007/s11565-009-0072-6. [12] R. Farwig and T. Hishida, Asymptotic profile of steady Stokes flow around a rotating obstacle,, Manuscripta Math., 136 (2011), 315. doi: 10.1007/s00229-011-0479-0. [13] R. Farwig and T. Hishida, Leading term at infinity of steady Navier-Stokes flow around a rotating obstacle,, Math. Nachr., 284 (2011), 2065. doi: 10.1002/mana.200910192. [14] R. Farwig, T. Hishida and D. Müller, $L^q$-theory of a singular "winding'' integral operator arising from fluid dynamics,, Pacific J. Math., 215 (2004), 297. doi: 10.2140/pjm.2004.215.297. [15] R. Farwig, M. Krbec and Š. Nečasová, A weighted $L^q$ approach to Stokes flow around a rotating body,, Ann. Univ. Ferrara, 54 (2008), 61. doi: 10.1007/s11565-008-0040-6. [16] R. Farwig, M. Krbec and Š. Nečasová, A weighted $L^q$-approach to Oseen flow around a rotating body,, Math. Meth. Appl. Sci., 31 (2008), 551. doi: 10.1002/mma.925. [17] R. Farwig and J. Neustupa, On the spectrum of a Stokes-type operator arising from flow around a rotating body,, Manuscripta Math., 122 (2007), 419. doi: 10.1007/s00229-007-0078-2. [18] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearised Steady Problems,, Springer Tracts in Natural Philosophy, (1998). doi: 10.1007/978-1-4612-5364-8. [19] G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications,, in Handbook of Mathematical Fluid Dynamics. Vol. I (eds. S. Friedlander and D. Serre), (2002), 653. [20] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems,, 2nd edition, (2011). doi: 10.1007/978-0-387-09620-9. [21] G. P. Galdi and M. Kyed, Steady-state Navier-Stokes flows past a rotating body: Leray solutions are physically reasonable,, Arch. Rat. Mech. Anal., 200 (2011), 21. doi: 10.1007/s00205-010-0350-6. [22] G. P. Galdi and M. Kyed, Asymptotic behavior of a Leray solution around a rotating obstacle,, Progress in Nonlinear Differential Equations and Their Applications, 60 (2011), 251. doi: 10.1007/978-3-0348-0075-4_13. [23] G. P. Galdi and M. Kyed, 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 (2013), 573. doi: 10.1090/S0002-9939-2012-11638-7. [24] G. P. Galdi and M. Kyed, 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 (2013), 1313. doi: 10.1090/S0002-9939-2012-11640-5. [25] G. P. Galdi and A. S. Silvestre, Strong solutions to the Navier-Stokes equations around a rotating obstacle,, Arch. Rat. Mech. Anal., 176 (2005), 331. doi: 10.1007/s00205-004-0348-z. [26] G. P. Galdi and S. A. Silvestre, The steady motion of a Navier-Stokes liquid around a rigid body,, Arch. Rat. Mech. Anal., 184 (2007), 371. doi: 10.1007/s00205-006-0026-4. [27] G. P. Galdi and S. A. Silvestre, Further results on steady-state flow of a Navier-Stokes liquid around a rigid body. Existence of the wake,, RIMS Kôkyûroku Bessatsu, B1 (2008), 108. [28] R. B. Guenther and E. A. Thomann, 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. doi: 10.1007/s00021-004-0139-1. [29] T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle,, Arch. Rat. Mech. Anal., 150 (1999), 307. doi: 10.1007/s002050050190. [30] T. Hishida, The Stokes operator with rotating effect in exterior domains,, Analysis, 19 (1999), 51. doi: 10.1524/anly.1999.19.1.51. [31] T. Hishida, $L^q$ estimates of weak solutions to the stationary Stokes equations around a rotating body,, J. Math. Soc. Japan, 58 (2006), 744. doi: 10.2969/jmsj/1156342036. [32] T. Hishida and Y. Shibata, $L_p$-$L_q$ estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle,, RIMS Kôkyûroku Bessatsu, B1 (2007), 167. [33] S. Kračmar, M. Krbec, Š. Nečasová, P. Penel and K. Schumacher, On the $L^q$-approach with generalized anisotropic weights of the weak solution of the Oseen flow around a rotating body,, Nonlinear Analysis, 71 (2009). doi: 10.1016/j.na.2009.07.017. [34] S. Kračmar, Š. Nečasová and P. Penel, Estimates of weak solutions in anisotropically weighted Sobolev spaces to the stationary rotating Oseen equations,, IASME Transactions, 2 (2005), 854. [35] S. Kračmar, Š. Nečasová and P. Penel, Anisotropic $L^2$ estimates of weak solutions to the stationary Oseen type equations in $\mathbbR ^{3}$ for a rotating body,, RIMS Kôkyûroku Bessatsu, B1 (2007), 219. [36] S. Kračmar, Š. Nečasová and P. Penel, Anisotropic $L^2$ 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. doi: 10.2969/jmsj/06210239. [37] S. Kračmar and P. Penel, Variational properties of a generic model equation in exterior 3D domains,, Funkcialaj Ekvacioj, 47 (2004), 499. doi: 10.1619/fesi.47.499. [38] S. Kračmar and P. Penel, New regularity results for a generic model equation in exterior 3D domains,, Banach Center Publications Warsaw, 70 (2005), 139. doi: 10.4064/bc70-0-9. [39] M. Kyed, Asymptotic profile of a linearized flow past a rotating body,, Quart. Appl. Math., 71 (2013), 489. doi: 10.1090/S0033-569X-2013-01288-7. [40] M. Kyed, On the asymptotic structure of a Navier-Stokes flow past a rotating body,, to appear in J. Math. Soc. Japan., (). doi: 10.2969/jmsj/06610001. [41] M. Kyed, On a mapping property of the Oseen operator with rotation,, Discrete Contin. Dynam. Syst. - Ser. S., 6 (2013), 1315. doi: 10.3934/dcdss.2013.6.1315. [42] Š. Nečasová, Asymptotic properties of the steady fall of a body in viscous fluids,, Math. Meth. Appl. Sci., 27 (2004), 1969. doi: 10.1002/mma.467. [43] Š. Nečasová, On the problem of the Stokes flow and Oseen flow in $\mathbbR^{3}$ with Coriolis force arising from fluid dynamics,, IASME Transaction, 2 (2005), 1262. [44] Š. Nečasová, K. Schumacher, Strong solution to the Stokes equations of a flow around a rotating body in weighted $L^q$ spaces,, Math. Nachr., 284 (2011), 1701. doi: 10.1002/mana.200810166.
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