# American Institute of Mathematical Sciences

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

 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

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.
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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated problems, Arch. Ration. Mech. Anal., 19 (1965), 363-406.  Google Scholar [19] A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall Inc., Englewood Cliffs, NJ, 1964.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] T. Hishida, The Stokes operator with rotating effect in exterior domains, Analysis (Munich), 19 (1999), 51-67.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar

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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), 237-253. doi: 10.3934/dcdss.2010.3.237.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated problems, Arch. Ration. Mech. Anal., 19 (1965), 363-406.  Google Scholar [19] A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall Inc., Englewood Cliffs, NJ, 1964.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] T. Hishida, The Stokes operator with rotating effect in exterior domains, Analysis (Munich), 19 (1999), 51-67.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar
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