# 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 - A, 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.  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.  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.  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, (2011), 351.   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.  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.  doi: 10.2748/tmj/1145390210.  Google Scholar [7] R. Farwig, Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle,, in, 70 (2005), 73.  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.  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.  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.  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.  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.  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.  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.  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, 54 (2008), 61.  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.  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.  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.   Google Scholar [19] A. Friedman, "Partial Differential Equations of Parabolic Type,", Prentice-Hall Inc., (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 (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, (2002), 653.   Google Scholar [22] G. P. Galdi, Steady flow of a Navier-Stokes fluid around a rotating obstacle,, J. Elasticity, 71 (2003), 1.  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, 80 (2011), 251.  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.  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.  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, (2007), 127.   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.  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.  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.   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.  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.  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.  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.   Google Scholar [34] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", Second English edition, (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, (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.  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.   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.  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.  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.  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.  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.  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, (2011), 351.   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.  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.  doi: 10.2748/tmj/1145390210.  Google Scholar [7] R. Farwig, Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle,, in, 70 (2005), 73.  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.  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.  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.  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.  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.  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.  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.  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, 54 (2008), 61.  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.  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.  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.   Google Scholar [19] A. Friedman, "Partial Differential Equations of Parabolic Type,", Prentice-Hall Inc., (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 (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, (2002), 653.   Google Scholar [22] G. P. Galdi, Steady flow of a Navier-Stokes fluid around a rotating obstacle,, J. Elasticity, 71 (2003), 1.  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, 80 (2011), 251.  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.  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.  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, (2007), 127.   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.  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.  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.   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.  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.  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.  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.   Google Scholar [34] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", Second English edition, (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, (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.  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.   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.  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.  doi: 10.1007/s00021-004-0139-1.  Google Scholar
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