# 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, (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.  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.  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, 35 (1999), 273.   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.  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.  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.  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.  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.  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.   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.   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.  doi: 10.1007/BF02366850.  Google Scholar [14] L. Nirenberg, On elliptic partial differential equations,, Ann. Sc. Norm. Super. Pisa (3), 13 (1959), 115.   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.   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, (1995), 63.   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.   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.  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.  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.  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, (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.  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.  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, 35 (1999), 273.   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.  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.  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.  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.  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.  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.   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.   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.  doi: 10.1007/BF02366850.  Google Scholar [14] L. Nirenberg, On elliptic partial differential equations,, Ann. Sc. Norm. Super. Pisa (3), 13 (1959), 115.   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.   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, (1995), 63.   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.   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.  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.  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.  doi: 10.1007/PL00004418.  Google Scholar
 [1] Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142 [2] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 [3] Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001 [4] Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 [5] Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234 [6] Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398 [7] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [8] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110 [9] Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020408 [10] Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160 [11] Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352 [12] Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM. Electronic Research Archive, , () : -. doi: 10.3934/era.2020128 [13] Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037 [14] Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations & Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039 [15] Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141 [16] Duy Phan. Approximate controllability for Navier–Stokes equations in $\rm3D$ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations & Control Theory, 2021, 10 (1) : 199-227. doi: 10.3934/eect.2020062 [17] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [18] Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229 [19] Michal Fečkan, Kui Liu, JinRong Wang. $(\omega,\mathbb{T})$-periodic solutions of impulsive evolution equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021006 [20] Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003

2019 Impact Factor: 1.233