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October  2013, 6(5): 1237-1257. doi: 10.3934/dcdss.2013.6.1237

Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane

Giovanni P. Galdi 1,

1. 

Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15261, United States

Received  November 2011 Revised  February 2012 Published  March 2013

We consider the two-dimensional motion of a Navier-Stokes liquid in the whole plane, under the action of a time-periodic body force $F$ of period $T$, and tending to a prescribed nonzero constant velocity at infinity. We show that if the magnitude of $F$, in suitable norm, is sufficiently small, there exists one and only one corresponding time-periodic flow of period $T$ in an appropriate function class.
Keywords: Navier-Stokes equations, time-periodic solutions, plane flow..
Mathematics Subject Classification: Primary: 35Q30, 76D; Secondary: 76.
Citation: Giovanni P. Galdi. Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1237-1257. doi: 10.3934/dcdss.2013.6.1237
References:
[1]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems," $2^{nd}$ edition, Springer Monographs in Mathematics, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[2]

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. Amer. Math. Soc., 141 (2013), 573-583. doi: 10.1090/S0002-9939-2012-11638-7.  Google Scholar

[3]

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. Amer. Math. Soc., 141 (2013), 1313-1322. doi: 10.1090/S0002-9939-2012-11640-5.  Google Scholar

[4]

G. P. Galdi and P. J. Rabier, Functional properties of the Navier-Stokes operator and bifurcation of stationary solutions: Planar exterior domains, in "Topics in Nonlinear Analysis," Progress in Nonlin. Diff. Equations Appl., 35, Birkhäuser, Basel, (1999) 273-303.  Google Scholar

[5]

G. P. Galdi and A. L. Silvestre, Existence of time-periodic solutions to the Navier-Stokes equations around a moving body, Pacific J. Math., 223 (2006), 251-267. doi: 10.2140/pjm.2006.223.251.  Google Scholar

[6]

G. P. Galdi and A. L. Silvestre, On the motion of a rigid body in a Navier-Stokes liquid under the action of a time-periodic force, Indiana Univ. Math. J., 58 (2009), 2805-2842. doi: 10.1512/iumj.2009.58.3758.  Google Scholar

[7]

G. P. Galdi and H. Sohr, Existence and uniqueness of time-periodic physically reasonable Navier-Stokes flow past a body, Arch. Ration. Mech. Anal., 172 (2004), 363-406. doi: 10.1007/s00205-004-0306-9.  Google Scholar

[8]

K. Kang, H. Miura and T.-P. Tsai, Asymptotics of small exterior Navier-Stokes flows with non-decaying boundary data, Comm. Partial Differential Equations, 37 (2012), 1717-1753. doi: 10.1080/03605302.2012.708082.  Google Scholar

[9]

M. Kyed, Existence and asymptotic properties of time-periodic solutions to the three dimensional Navier-Stokes equations,, in preparation., ().   Google Scholar

[10]

H. Kozono and M. Nakao, Periodic solutions of the Navier-Stokes equations in unbounded domains, Tohoku Math. J. (2), 48 (1996), 33-50. doi: 10.2748/tmj/1178225411.  Google Scholar

[11]

J.-L. Lions, Espaces intermédiaires entre espaces hilbertiens et applications, (French) [Interpolation spaces between Hilbert spaces and applications], Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N.S.), 2(50) (1958), 419-432.  Google Scholar

[12]

P. Maremonti, Existence and stability of time periodic solutions of the Navier-Stokes equations in the whole space, Nonlinearity, 4 (1991), 503-529.  Google Scholar

[13]

P. Maremonti and M. Padula, Existence, uniqueness and attainability of periodic solutions of the Navier-Stokes equations in exterior domains, J. Math. Sci. (New York), 93 (1999), 719-746. doi: 10.1007/BF02366850.  Google Scholar

[14]

L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa (3), 13 (1959), 115-162.  Google Scholar

[15]

G. Prodi, Teoremi di tipo locale per il sistema di Navier-Stokes e stabilità delle soluzioni stazionarie, (Italian) [Local theorems for the Navier-Stokes system and stability of steady-state solutions], Rend. Sem. Mat. Univ. Padova, 32 (1962), 374-397.  Google Scholar

[16]

R. Salvi, On the existence of periodic weak solutions on the Navier-Stokes equations in exterior regions with periodically moving boundaries, in "Navier-Stokes Equations and Related Nonlinear Problems'' (ed. A. Sequeira) (Funchal, 1994), Plenum, New York, (1995), 63-73.  Google Scholar

[17]

V. A. Solonnikov, Estimates of the solutions of the nonstationary Navier-Stokes system, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 153-231.  Google Scholar

[18]

Y. Taniuchi, On the uniqueness of time-periodic solutions to the Navier-Stokes equations in unbounded domains, Math. Z., 261 (2009), 597-615. doi: 10.1007/s00209-008-0341-6.  Google Scholar

[19]

G. van Baalen and P. Wittwer, Time periodic solutions of the Navier-Stokes equations with nonzero constant boundary conditions at infinity, SIAM J. Math. Anal., 43 (2011), 1787-1809. doi: 10.1137/100809842.  Google Scholar

[20]

M. Yamazaki, The Navier-Stokes equations in the weak$-L^n$ space with time-dependent external force, Math. Ann., 317 (2000), 635-675. doi: 10.1007/PL00004418.  Google Scholar

show all references

References:
[1]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems," $2^{nd}$ edition, Springer Monographs in Mathematics, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[2]

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. Amer. Math. Soc., 141 (2013), 573-583. doi: 10.1090/S0002-9939-2012-11638-7.  Google Scholar

[3]

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. Amer. Math. Soc., 141 (2013), 1313-1322. doi: 10.1090/S0002-9939-2012-11640-5.  Google Scholar

[4]

G. P. Galdi and P. J. Rabier, Functional properties of the Navier-Stokes operator and bifurcation of stationary solutions: Planar exterior domains, in "Topics in Nonlinear Analysis," Progress in Nonlin. Diff. Equations Appl., 35, Birkhäuser, Basel, (1999) 273-303.  Google Scholar

[5]

G. P. Galdi and A. L. Silvestre, Existence of time-periodic solutions to the Navier-Stokes equations around a moving body, Pacific J. Math., 223 (2006), 251-267. doi: 10.2140/pjm.2006.223.251.  Google Scholar

[6]

G. P. Galdi and A. L. Silvestre, On the motion of a rigid body in a Navier-Stokes liquid under the action of a time-periodic force, Indiana Univ. Math. J., 58 (2009), 2805-2842. doi: 10.1512/iumj.2009.58.3758.  Google Scholar

[7]

G. P. Galdi and H. Sohr, Existence and uniqueness of time-periodic physically reasonable Navier-Stokes flow past a body, Arch. Ration. Mech. Anal., 172 (2004), 363-406. doi: 10.1007/s00205-004-0306-9.  Google Scholar

[8]

K. Kang, H. Miura and T.-P. Tsai, Asymptotics of small exterior Navier-Stokes flows with non-decaying boundary data, Comm. Partial Differential Equations, 37 (2012), 1717-1753. doi: 10.1080/03605302.2012.708082.  Google Scholar

[9]

M. Kyed, Existence and asymptotic properties of time-periodic solutions to the three dimensional Navier-Stokes equations,, in preparation., ().   Google Scholar

[10]

H. Kozono and M. Nakao, Periodic solutions of the Navier-Stokes equations in unbounded domains, Tohoku Math. J. (2), 48 (1996), 33-50. doi: 10.2748/tmj/1178225411.  Google Scholar

[11]

J.-L. Lions, Espaces intermédiaires entre espaces hilbertiens et applications, (French) [Interpolation spaces between Hilbert spaces and applications], Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N.S.), 2(50) (1958), 419-432.  Google Scholar

[12]

P. Maremonti, Existence and stability of time periodic solutions of the Navier-Stokes equations in the whole space, Nonlinearity, 4 (1991), 503-529.  Google Scholar

[13]

P. Maremonti and M. Padula, Existence, uniqueness and attainability of periodic solutions of the Navier-Stokes equations in exterior domains, J. Math. Sci. (New York), 93 (1999), 719-746. doi: 10.1007/BF02366850.  Google Scholar

[14]

L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa (3), 13 (1959), 115-162.  Google Scholar

[15]

G. Prodi, Teoremi di tipo locale per il sistema di Navier-Stokes e stabilità delle soluzioni stazionarie, (Italian) [Local theorems for the Navier-Stokes system and stability of steady-state solutions], Rend. Sem. Mat. Univ. Padova, 32 (1962), 374-397.  Google Scholar

[16]

R. Salvi, On the existence of periodic weak solutions on the Navier-Stokes equations in exterior regions with periodically moving boundaries, in "Navier-Stokes Equations and Related Nonlinear Problems'' (ed. A. Sequeira) (Funchal, 1994), Plenum, New York, (1995), 63-73.  Google Scholar

[17]

V. A. Solonnikov, Estimates of the solutions of the nonstationary Navier-Stokes system, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 153-231.  Google Scholar

[18]

Y. Taniuchi, On the uniqueness of time-periodic solutions to the Navier-Stokes equations in unbounded domains, Math. Z., 261 (2009), 597-615. doi: 10.1007/s00209-008-0341-6.  Google Scholar

[19]

G. van Baalen and P. Wittwer, Time periodic solutions of the Navier-Stokes equations with nonzero constant boundary conditions at infinity, SIAM J. Math. Anal., 43 (2011), 1787-1809. doi: 10.1137/100809842.  Google Scholar

[20]

M. Yamazaki, The Navier-Stokes equations in the weak$-L^n$ space with time-dependent external force, Math. Ann., 317 (2000), 635-675. doi: 10.1007/PL00004418.  Google Scholar

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