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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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
M. Kyed, Existence and asymptotic properties of time-periodic solutions to the three dimensional Navier-Stokes equations, in preparation. |
[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. |
[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. |
[12] |
P. Maremonti, Existence and stability of time periodic solutions of the Navier-Stokes equations in the whole space, Nonlinearity, 4 (1991), 503-529. |
[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. |
[14] |
L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa (3), 13 (1959), 115-162. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
M. Kyed, Existence and asymptotic properties of time-periodic solutions to the three dimensional Navier-Stokes equations, in preparation. |
[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. |
[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. |
[12] |
P. Maremonti, Existence and stability of time periodic solutions of the Navier-Stokes equations in the whole space, Nonlinearity, 4 (1991), 503-529. |
[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. |
[14] |
L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa (3), 13 (1959), 115-162. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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