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

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.  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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