# American Institute of Mathematical Sciences

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

 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.
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

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##### 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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