The fundamental solution of linearized nonstationary Navier-Stokes equations of motion around a rotating and translating body

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February 2014, 34(2): 511-529. doi: 10.3934/dcds.2014.34.511

The fundamental solution of linearized nonstationary Navier-Stokes equations of motion around a rotating and translating body

Reinhard Farwig ^1, , Ronald B. Guenther ^2, , Enrique A. Thomann ^2, and Šárka Nečasová ^3,

1.	Department of Mathematics and Center of Smart Interfaces (CSI), Technische Universität Darmstadt, 64289 Darmstadt, Germany
2.	Department of Mathematics, Oregon State University, Corvallis, OR 97331, United States, United States
3.	Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1

Received July 2012 Revised April 2013 Published August 2013

Abstract
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We derive the fundamental solution of the linearized problem of the motion of a viscous fluid around a rotating body when the axis of rotation of the body is not parallel to the velocity of the fluid at infinity.

Keywords: Navier-Stokes problem, linearized problem, rotating body, Fundamental solution, wake., translating body.

Mathematics Subject Classification: Primary: 35Q30, 76D07; Secondary: 35A08, 35Q35, 76D0.

Citation: Reinhard Farwig, Ronald B. Guenther, Enrique A. Thomann, Šárka Nečasová. The fundamental solution of linearized nonstationary Navier-Stokes equations of motion around a rotating and translating body. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 511-529. doi: 10.3934/dcds.2014.34.511

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, Dynamical Systems, Differential Equations and Applications, $8^{th}$ AIMS Conference, Supplement Vol. I, (2011), 351-361.
[5]	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.
[6]	R. Farwig, An $L^q$-analysis of viscous fluid flow past a rotating obstacle, Tôhoku Math. J., 58 (2006), 129-147. doi: 10.2748/tmj/1145390210.
[7]	R. Farwig, Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle, in "Regularity and Other Aspects of the Navier-Stokes Equations," Banach Center Publ., 70, Polish Acad. Sci., Warsaw, (2005), 73-84. doi: 10.4064/bc70-0-5.
[8]	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.
[9]	R. Farwig and T. Hishida, Stationary Navier-Stokes flow around a rotating obstacles, Funkcial. Ekvac., 50 (2007), 371-403. doi: 10.1619/fesi.50.371.
[10]	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.
[11]	R. Farwig and T. Hishida, Asymptotic profiles of steady Stokes and Navier-Stokes flows around a rotating obstacle, Ann. Univ. Ferrara Sez. VII Sci. Mat., 55 (2009), 263-277. doi: 10.1007/s11565-009-0072-6.
[12]	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.
[13]	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.
[14]	R. Farwig, M. Krbec and Š. Nečasová, A weighted $L^q$-approach to Oseen flow around a rotating body, Math. Methods Appl. Sci., 31 (2008), 551-574. doi: 10.1002/mma.925.
[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 Sci. Mat., 54 (2008), 61-84. doi: 10.1007/s11565-008-0040-6.
[16]	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.
[17]	R. Farwig, Š. Nečasová and J. Neustupa, Spectral analysis of a Stokes-type operator arising from flow around a rotating body, J. Math. Soc. Japan, 63 (2011), 163-194. doi: 10.2969/jmsj/06310163.
[18]	R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated problems, Arch. Ration. Mech. Anal., 19 (1965), 363-406.
[19]	A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall Inc., Englewood Cliffs, NJ, 1964.
[20]	G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems," Springer Tracts in Natural Philosophy, 38, Springer-Verlag, New York 1994. doi: 10.1007/978-1-4612-5364-8.
[21]	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. 1" (eds. S. Friedlander and D. Serre), North-Holland, Amsterdam, (2002), 653-791.
[22]	G. P. Galdi, Steady flow of a Navier-Stokes fluid around a rotating obstacle, J. Elasticity, 71 (2003), 1-31. doi: 10.1023/B:ELAS.0000005543.00407.5e.
[23]	G. P. Galdi and M. Kyed, Asymptotic behavior of a Leray solution around a rotating obstacle, in "Parabolic Problems: The Herbert Amann Festschrift" (eds. J. Escher, et al.), Prog. Nonl. Diff. Eq. Appl., 80, Springer, Basel, (2011), 251-266. doi: 10.1007/978-3-0348-0075-4_13.
[24]	G. P. Galdi and M. Kyed, Steady-state Navier-Stokes flows past a rotating body: Leray solutions are physically reasonable, Arch. Ration. Mech. Anal., 200 (2011), 21-58. doi: 10.1007/s00205-010-0350-6.
[25]	G. P. Galdi and A. L. Silvestre, On the steady motion of a Navier-Stokes liquid around a rigid body, Arch. Rational Mech. Anal., 184 (2006), 371-400. doi: 10.1007/s00205-006-0026-4.
[26]	G. P. Galdi and A. L. Silvestre, Further results on steady-state flow of a Navier-Stokes liquid around a rigid body. Existence of the wake, in "Kyoto Conference on the Navier-Stokes Equations and their Applications," RIMS Kôkyûroku Bessatsu, B1, Res. Inst. Math. Sci. (RIMS), Kyoto, (2007), 127-143.
[27]	M. Geissert, H. Heck and M. Hieber, $L^p$ theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle, J. Reine Angew. Math., 596 (2006), 45-62. doi: 10.1515/CRELLE.2006.051.
[28]	T. Hansel, On the Navier-Stokes equations with rotating effect and prescribed outflow velocity, J. Math. Fluid Mech., 13 (2011), 405-419. doi: 10.1007/s00021-010-0026-x.
[29]	T. Hishida, The Stokes operator with rotating effect in exterior domains, Analysis (Munich), 19 (1999), 51-67.
[30]	T. Hishida, $L^q$ estimates of weak solutions to the stationary Stokes equations around a rotating body, J. Math. Soc. Japan, 58 (2006), 743-767. doi: 10.2969/jmsj/1156342036.
[31]	T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Ration. Mech. Anal., 150 (1999), 307-348. doi: 10.1007/s002050050190.
[32]	T. Hishida and Y. Shibata, $L_p-L_q$ estimate of Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle, Arch. Ration. Mech. Anal., 193 (2009), 339-421. doi: 10.1007/s00205-008-0130-8.
[33]	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 Kokyuroku Bessatsu, B1 (2007), 219-235.
[34]	O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Second English edition, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.
[35]	W. Magnus, F. Oberhettinger and R. P. Soni, "Formulas and Theorems for the Special Functions of Mathematical Physics," Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966.
[36]	G. Da Prato and A. Lunardi, On the Ornstein-Uhlenbeck operator in spaces of continuous functions, J. Funct. Anal., 131 (1995), 94-114. doi: 10.1006/jfan.1995.1084.
[37]	V. A. Solonnikov, Estimates of the solutions of a nonstationary linearized system of Navier-Stokes equations, (English) Amer. Math. Soc. Transl. Ser. 2, 75 (1968), 1-116.
[38]	V. A. Solonnikov, On estimates of solutions of the non-stationary Stokes problem in anisotropic Sobolev spaces and on estimates for the resolvent of the Stokes operator, (English) Russian Math. Surveys, 58 (2003), 331-365; (Russian) Uspekhi Mat. Nauk, 58 (2003), 123-156. doi: 10.1070/RM2003v058n02ABEH000613.
[39]	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. doi: 10.1007/s00021-004-0139-1.

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, Dynamical Systems, Differential Equations and Applications, $8^{th}$ AIMS Conference, Supplement Vol. I, (2011), 351-361.
[5]	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.
[6]	R. Farwig, An $L^q$-analysis of viscous fluid flow past a rotating obstacle, Tôhoku Math. J., 58 (2006), 129-147. doi: 10.2748/tmj/1145390210.
[7]	R. Farwig, Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle, in "Regularity and Other Aspects of the Navier-Stokes Equations," Banach Center Publ., 70, Polish Acad. Sci., Warsaw, (2005), 73-84. doi: 10.4064/bc70-0-5.
[8]	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.
[9]	R. Farwig and T. Hishida, Stationary Navier-Stokes flow around a rotating obstacles, Funkcial. Ekvac., 50 (2007), 371-403. doi: 10.1619/fesi.50.371.
[10]	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.
[11]	R. Farwig and T. Hishida, Asymptotic profiles of steady Stokes and Navier-Stokes flows around a rotating obstacle, Ann. Univ. Ferrara Sez. VII Sci. Mat., 55 (2009), 263-277. doi: 10.1007/s11565-009-0072-6.
[12]	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.
[13]	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.
[14]	R. Farwig, M. Krbec and Š. Nečasová, A weighted $L^q$-approach to Oseen flow around a rotating body, Math. Methods Appl. Sci., 31 (2008), 551-574. doi: 10.1002/mma.925.
[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 Sci. Mat., 54 (2008), 61-84. doi: 10.1007/s11565-008-0040-6.
[16]	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.
[17]	R. Farwig, Š. Nečasová and J. Neustupa, Spectral analysis of a Stokes-type operator arising from flow around a rotating body, J. Math. Soc. Japan, 63 (2011), 163-194. doi: 10.2969/jmsj/06310163.
[18]	R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated problems, Arch. Ration. Mech. Anal., 19 (1965), 363-406.
[19]	A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall Inc., Englewood Cliffs, NJ, 1964.
[20]	G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems," Springer Tracts in Natural Philosophy, 38, Springer-Verlag, New York 1994. doi: 10.1007/978-1-4612-5364-8.
[21]	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. 1" (eds. S. Friedlander and D. Serre), North-Holland, Amsterdam, (2002), 653-791.
[22]	G. P. Galdi, Steady flow of a Navier-Stokes fluid around a rotating obstacle, J. Elasticity, 71 (2003), 1-31. doi: 10.1023/B:ELAS.0000005543.00407.5e.
[23]	G. P. Galdi and M. Kyed, Asymptotic behavior of a Leray solution around a rotating obstacle, in "Parabolic Problems: The Herbert Amann Festschrift" (eds. J. Escher, et al.), Prog. Nonl. Diff. Eq. Appl., 80, Springer, Basel, (2011), 251-266. doi: 10.1007/978-3-0348-0075-4_13.
[24]	G. P. Galdi and M. Kyed, Steady-state Navier-Stokes flows past a rotating body: Leray solutions are physically reasonable, Arch. Ration. Mech. Anal., 200 (2011), 21-58. doi: 10.1007/s00205-010-0350-6.
[25]	G. P. Galdi and A. L. Silvestre, On the steady motion of a Navier-Stokes liquid around a rigid body, Arch. Rational Mech. Anal., 184 (2006), 371-400. doi: 10.1007/s00205-006-0026-4.
[26]	G. P. Galdi and A. L. Silvestre, Further results on steady-state flow of a Navier-Stokes liquid around a rigid body. Existence of the wake, in "Kyoto Conference on the Navier-Stokes Equations and their Applications," RIMS Kôkyûroku Bessatsu, B1, Res. Inst. Math. Sci. (RIMS), Kyoto, (2007), 127-143.
[27]	M. Geissert, H. Heck and M. Hieber, $L^p$ theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle, J. Reine Angew. Math., 596 (2006), 45-62. doi: 10.1515/CRELLE.2006.051.
[28]	T. Hansel, On the Navier-Stokes equations with rotating effect and prescribed outflow velocity, J. Math. Fluid Mech., 13 (2011), 405-419. doi: 10.1007/s00021-010-0026-x.
[29]	T. Hishida, The Stokes operator with rotating effect in exterior domains, Analysis (Munich), 19 (1999), 51-67.
[30]	T. Hishida, $L^q$ estimates of weak solutions to the stationary Stokes equations around a rotating body, J. Math. Soc. Japan, 58 (2006), 743-767. doi: 10.2969/jmsj/1156342036.
[31]	T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Ration. Mech. Anal., 150 (1999), 307-348. doi: 10.1007/s002050050190.
[32]	T. Hishida and Y. Shibata, $L_p-L_q$ estimate of Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle, Arch. Ration. Mech. Anal., 193 (2009), 339-421. doi: 10.1007/s00205-008-0130-8.
[33]	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 Kokyuroku Bessatsu, B1 (2007), 219-235.
[34]	O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Second English edition, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.
[35]	W. Magnus, F. Oberhettinger and R. P. Soni, "Formulas and Theorems for the Special Functions of Mathematical Physics," Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966.
[36]	G. Da Prato and A. Lunardi, On the Ornstein-Uhlenbeck operator in spaces of continuous functions, J. Funct. Anal., 131 (1995), 94-114. doi: 10.1006/jfan.1995.1084.
[37]	V. A. Solonnikov, Estimates of the solutions of a nonstationary linearized system of Navier-Stokes equations, (English) Amer. Math. Soc. Transl. Ser. 2, 75 (1968), 1-116.
[38]	V. A. Solonnikov, On estimates of solutions of the non-stationary Stokes problem in anisotropic Sobolev spaces and on estimates for the resolvent of the Stokes operator, (English) Russian Math. Surveys, 58 (2003), 331-365; (Russian) Uspekhi Mat. Nauk, 58 (2003), 123-156. doi: 10.1070/RM2003v058n02ABEH000613.
[39]	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. doi: 10.1007/s00021-004-0139-1.

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