# American Institute of Mathematical Sciences

July  2014, 13(4): 1361-1393. doi: 10.3934/cpaa.2014.13.1361

## Infinite-energy solutions for the Navier-Stokes equations in a strip revisited

 1 University of Surrey, Guildford, Gu27XH, Surrey, United Kingdom 2 Department of Mathematics, University of Surrey, Guildford, GU2 7XH

Received  October 2013 Revised  January 2014 Published  February 2014

The paper deals with the Navier-Stokes equations in a strip in the class of spatially non-decaying (in nite-energy) solutions belonging to the properly chosen uniformly local Sobolev spaces. The global well-posedness and dissipativity of the Navier-Stokes equations in a strip in such spaces has been rst established in [22]. However, the proof given there contains a rather essential error and the aim of the present paper is to correct this error and to show that the main results of [22] remain true.
Citation: Peter Anthony, Sergey Zelik. Infinite-energy solutions for the Navier-Stokes equations in a strip revisited. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1361-1393. doi: 10.3934/cpaa.2014.13.1361
##### References:
 [1] F. Abergel, Attractor for a Navier-Stokes flow in an unbounded domain. Attractors, inertial manifolds and their approximation (Marseille-Luminy, 1987)., \emph{RAIRO Model. Math. Anal. Numer.}, 23 (1989), 359. Google Scholar [2] F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains,, \emph{J. Differential Equations}, 83 (1990), 85. doi: 10.1016/0022-0396(90)90070-6. Google Scholar [3] A. Afendikov and A. Mielke, Dynamical properties of spatially non-decaying 2D Navier-Stokes flows with Kolmogorov forcing in an infinite strip,, \emph{J. Math. Fluid Mech.}, 7 (2005), 51. doi: 10.1007/s00021-004-0131-9. Google Scholar [4] H. Amann, On the strong solvability of the Navier-Stokes equations,, \emph{Jour. Math.Fluid Mechanics}, 2 (2000), 16. doi: 10.1007/s000210050018. Google Scholar [5] A. Babin, Asymptotic Expansions at infinity of a strongly perturbed Poiseuille flow,, \emph{Advances in Soviet Math.}, 10 (1992), 1. Google Scholar [6] A. Babin, The attractor of a Navier-Stokes system in an unbounded channel-like domain,, \emph{J. Dynam. Differential Equations}, 4 (1992), 555. doi: 10.1007/BF01048260. Google Scholar [7] A. Babin and M.Vishik, Attractors of partial differential evolution equations in an unbounded domain,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 116 (1990), 221. doi: 10.1017/S0308210500031498. Google Scholar [8] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Nauka, (1989). Google Scholar [9] M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain,, \emph{Comm. Pure Appl. Math.}, 54 (2001), 625. doi: 10.1002/cpa.1011. Google Scholar [10] Y. Giga, S. Matsui and O. Sawada, Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity,, \emph{J. Math. Fluid Mech.}, 3 (2001), 302. doi: 10.1007/PL00000973. Google Scholar [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). Google Scholar [12] P. Lemarie-Rieusset, Recent developments in the Navier-Stokes problem,, Chapman $&$ Hall/CRC Research Notes in Mathematics, (2002). doi: 10.1201/9781420035674. Google Scholar [13] A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains - existence and comparison,, \emph{Nonlinearity}, 8 (1995), 743. Google Scholar [14] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in \emph{Handbook of Differential Equations: Evolutionary Equations}, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar [15] J. Pennant and S. Zelik, Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\R^3$,, \emph{Comm. Pure Appl. Anal.}, 12 (2013), 461. Google Scholar [16] S. Revina and V. Yudovich, $L^p$-estimates for the resolvent of the Stokes operator in an infinite cylinder,, \emph{(Russian) Mat. Sb.}, 187 (1996), 97. doi: 10.1070/SM1996v187n06ABEH000139. Google Scholar [17] O. Sawada and Y. Taniuchi, A remark on $L^\infty$-solutions to the 2D Navier-Stokes equations,, \emph{J. Math. Fluid Mech.}, 9 (2007), 533. doi: 10.1007/s00021-005-0212-4. Google Scholar [18] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, North-Holland, (1977). Google Scholar [19] R. Temam, Infnite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematics Series, (1988). Google Scholar [20] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978). Google Scholar [21] S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip,, \emph{Glasg. Math. J.}, 49 (2007), 525. doi: 10.1017/S0017089507003849. Google Scholar [22] S. Zelik, Weak spatially nondecaying solutions of 3D Navier-Stokes equations in cylindrical domains,, Instability in models connected with fluid flows. II, (2008), 255. doi: 10.1007/978-0-387-75219-8_6. Google Scholar [23] S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity,, \emph{Comm. Pure Appl. Math.}, 56 (2003), 584. doi: 10.1002/cpa.10068. Google Scholar [24] S. Zelik, Infinite energy solutions for damped Navier-Stokes equations in $\R^2$,, \emph{Jour. Math. Fluid Mech.}, 15 (2013), 717. doi: 10.1007/s00021-013-0144-3. Google Scholar

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##### References:
 [1] F. Abergel, Attractor for a Navier-Stokes flow in an unbounded domain. Attractors, inertial manifolds and their approximation (Marseille-Luminy, 1987)., \emph{RAIRO Model. Math. Anal. Numer.}, 23 (1989), 359. Google Scholar [2] F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains,, \emph{J. Differential Equations}, 83 (1990), 85. doi: 10.1016/0022-0396(90)90070-6. Google Scholar [3] A. Afendikov and A. Mielke, Dynamical properties of spatially non-decaying 2D Navier-Stokes flows with Kolmogorov forcing in an infinite strip,, \emph{J. Math. Fluid Mech.}, 7 (2005), 51. doi: 10.1007/s00021-004-0131-9. Google Scholar [4] H. Amann, On the strong solvability of the Navier-Stokes equations,, \emph{Jour. Math.Fluid Mechanics}, 2 (2000), 16. doi: 10.1007/s000210050018. Google Scholar [5] A. Babin, Asymptotic Expansions at infinity of a strongly perturbed Poiseuille flow,, \emph{Advances in Soviet Math.}, 10 (1992), 1. Google Scholar [6] A. Babin, The attractor of a Navier-Stokes system in an unbounded channel-like domain,, \emph{J. Dynam. Differential Equations}, 4 (1992), 555. doi: 10.1007/BF01048260. Google Scholar [7] A. Babin and M.Vishik, Attractors of partial differential evolution equations in an unbounded domain,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 116 (1990), 221. doi: 10.1017/S0308210500031498. Google Scholar [8] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Nauka, (1989). Google Scholar [9] M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain,, \emph{Comm. Pure Appl. Math.}, 54 (2001), 625. doi: 10.1002/cpa.1011. Google Scholar [10] Y. Giga, S. Matsui and O. Sawada, Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity,, \emph{J. Math. Fluid Mech.}, 3 (2001), 302. doi: 10.1007/PL00000973. Google Scholar [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). Google Scholar [12] P. Lemarie-Rieusset, Recent developments in the Navier-Stokes problem,, Chapman $&$ Hall/CRC Research Notes in Mathematics, (2002). doi: 10.1201/9781420035674. Google Scholar [13] A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains - existence and comparison,, \emph{Nonlinearity}, 8 (1995), 743. Google Scholar [14] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in \emph{Handbook of Differential Equations: Evolutionary Equations}, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar [15] J. Pennant and S. Zelik, Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\R^3$,, \emph{Comm. Pure Appl. Anal.}, 12 (2013), 461. Google Scholar [16] S. Revina and V. Yudovich, $L^p$-estimates for the resolvent of the Stokes operator in an infinite cylinder,, \emph{(Russian) Mat. Sb.}, 187 (1996), 97. doi: 10.1070/SM1996v187n06ABEH000139. Google Scholar [17] O. Sawada and Y. Taniuchi, A remark on $L^\infty$-solutions to the 2D Navier-Stokes equations,, \emph{J. Math. Fluid Mech.}, 9 (2007), 533. doi: 10.1007/s00021-005-0212-4. Google Scholar [18] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, North-Holland, (1977). Google Scholar [19] R. Temam, Infnite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematics Series, (1988). Google Scholar [20] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978). Google Scholar [21] S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip,, \emph{Glasg. Math. J.}, 49 (2007), 525. doi: 10.1017/S0017089507003849. Google Scholar [22] S. Zelik, Weak spatially nondecaying solutions of 3D Navier-Stokes equations in cylindrical domains,, Instability in models connected with fluid flows. II, (2008), 255. doi: 10.1007/978-0-387-75219-8_6. Google Scholar [23] S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity,, \emph{Comm. Pure Appl. Math.}, 56 (2003), 584. doi: 10.1002/cpa.10068. Google Scholar [24] S. Zelik, Infinite energy solutions for damped Navier-Stokes equations in $\R^2$,, \emph{Jour. Math. Fluid Mech.}, 15 (2013), 717. doi: 10.1007/s00021-013-0144-3. Google Scholar
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