October  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 and 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), 237-253. 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), 705-738. 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), 459-476. 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, Dynamical Systems, Differential Equations and Applications, 8th AIMS Conference, Suppl. Vol. I, 351-361.

[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), 1576-1606. 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), 409-447. doi: 10.1007/BF02571437.

[7]

R. Farwig, An $L^q$-analysis of viscous fluid flow past a rotating obstacle, Tôhoku Math. J., 58 (2006), 1-147. 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), 73-84. 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), 367-382. 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), 371-403. 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, Sez. VII, 55 (2009), 263-277. 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), 315-338. 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), 2065-2077 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), 297-312. 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, Sez. VII, 54 (2008), 61-84. 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), 551-574. 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), 419-437. 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, 38, Springer, New York e.a., 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), North-Holland, Amsterdam, 2002, 653-791.

[20]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2nd edition, Springer, New York e.a., 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), 21-58. 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), 251-266. 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), 573-583. 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), 1313-1322. 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), 331-350. 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), 371-400. 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), 108-127.

[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), 77-98. 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), 307-348. doi: 10.1007/s002050050190.

[30]

T. Hishida, The Stokes operator with rotating effect in exterior domains, Analysis, 19 (1999), 51-67. 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), 744-767. 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), 167-188.

[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), e2940-e2957. 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), 854-861.

[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), 219-235.

[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), 239-268. 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), 499-523. 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), 139-155. doi: 10.4064/bc70-0-9.

[39]

M. Kyed, Asymptotic profile of a linearized flow past a rotating body, Quart. Appl. Math., 71 (2013), 489-500. 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), 1315-1322. 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), 1969-1995. 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), 1262-1270.

[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), 1701-1714. doi: 10.1002/mana.200810166.

show all references

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), 237-253. 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), 705-738. 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), 459-476. 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, Dynamical Systems, Differential Equations and Applications, 8th AIMS Conference, Suppl. Vol. I, 351-361.

[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), 1576-1606. 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), 409-447. doi: 10.1007/BF02571437.

[7]

R. Farwig, An $L^q$-analysis of viscous fluid flow past a rotating obstacle, Tôhoku Math. J., 58 (2006), 1-147. 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), 73-84. 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), 367-382. 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), 371-403. 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, Sez. VII, 55 (2009), 263-277. 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), 315-338. 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), 2065-2077 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), 297-312. 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, Sez. VII, 54 (2008), 61-84. 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), 551-574. 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), 419-437. 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, 38, Springer, New York e.a., 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), North-Holland, Amsterdam, 2002, 653-791.

[20]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2nd edition, Springer, New York e.a., 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), 21-58. 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), 251-266. 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), 573-583. 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), 1313-1322. 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), 331-350. 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), 371-400. 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), 108-127.

[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), 77-98. 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), 307-348. doi: 10.1007/s002050050190.

[30]

T. Hishida, The Stokes operator with rotating effect in exterior domains, Analysis, 19 (1999), 51-67. 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), 744-767. 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), 167-188.

[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), e2940-e2957. 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), 854-861.

[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), 219-235.

[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), 239-268. 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), 499-523. 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), 139-155. doi: 10.4064/bc70-0-9.

[39]

M. Kyed, Asymptotic profile of a linearized flow past a rotating body, Quart. Appl. Math., 71 (2013), 489-500. 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), 1315-1322. 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), 1969-1995. 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), 1262-1270.

[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), 1701-1714. doi: 10.1002/mana.200810166.

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