# 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 and 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). RAIRO Model. Math. Anal. Numer., 23 (1989), 359-370. [2] F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations, 83 (1990), 85-108. doi: 10.1016/0022-0396(90)90070-6. [3] A. Afendikov and A. Mielke, Dynamical properties of spatially non-decaying 2D Navier-Stokes flows with Kolmogorov forcing in an infinite strip, J. Math. Fluid Mech., 7 (2005), 51-67. doi: 10.1007/s00021-004-0131-9. [4] H. Amann, On the strong solvability of the Navier-Stokes equations, Jour. Math.Fluid Mechanics, 2 (2000), 16-98. doi: 10.1007/s000210050018. [5] A. Babin, Asymptotic Expansions at infinity of a strongly perturbed Poiseuille flow, Advances in Soviet Math., 10 (1992), 1-83. [6] A. Babin, The attractor of a Navier-Stokes system in an unbounded channel-like domain, J. Dynam. Differential Equations, 4 (1992), 555-584. doi: 10.1007/BF01048260. [7] A. Babin and M.Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243. doi: 10.1017/S0308210500031498. [8] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989; North Holland, Amsterdam, 1992. [9] M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688. doi: 10.1002/cpa.1011. [10] Y. Giga, S. Matsui and O. Sawada, Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity, J. Math. Fluid Mech., 3 (2001), 302-315. doi: 10.1007/PL00000973. [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. [12] P. Lemarie-Rieusset, Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, 431. Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674. [13] A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains - existence and comparison, Nonlinearity, 8 (1995), 743-768. [14] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations, Vol. IV, 103-200, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0. [15] J. Pennant and S. Zelik, Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $mathbb{R}^{3}$, Comm. Pure Appl. Anal., 12 (2013), 461-480. [16] S. Revina and V. Yudovich, $L^p$-estimates for the resolvent of the Stokes operator in an infinite cylinder, (Russian) Mat. Sb., 187 (1996), 97-118; translation in Sb. Math., 187 (1996), 881-902. doi: 10.1070/SM1996v187n06ABEH000139. [17] O. Sawada and Y. Taniuchi, A remark on $L^\infty$-solutions to the 2D Navier-Stokes equations, J. Math. Fluid Mech., 9 (2007), 533-542. doi: 10.1007/s00021-005-0212-4. [18] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam New York-Oxford, 1977. [19] R. Temam, Infnite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Series, Springer, New York-Berlin, 1988; 2nd ed., New York, 1997. [20] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978. [21] S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip, Glasg. Math. J., 49 (2007), 525-588. doi: 10.1017/S0017089507003849. [22] S. Zelik, Weak spatially nondecaying solutions of 3D Navier-Stokes equations in cylindrical domains, Instability in models connected with fluid flows. II, 255-327, Int. Math. Ser. (N. Y.), 7, Springer, New York, 2008. doi: 10.1007/978-0-387-75219-8_6. [23] S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math., 56 (2003), 584-637. doi: 10.1002/cpa.10068. [24] S. Zelik, Infinite energy solutions for damped Navier-Stokes equations in $\mathbb{R}^2$, Jour. Math. Fluid Mech., 15 (2013), 717-745. doi: 10.1007/s00021-013-0144-3.

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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). RAIRO Model. Math. Anal. Numer., 23 (1989), 359-370. [2] F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations, 83 (1990), 85-108. doi: 10.1016/0022-0396(90)90070-6. [3] A. Afendikov and A. Mielke, Dynamical properties of spatially non-decaying 2D Navier-Stokes flows with Kolmogorov forcing in an infinite strip, J. Math. Fluid Mech., 7 (2005), 51-67. doi: 10.1007/s00021-004-0131-9. [4] H. Amann, On the strong solvability of the Navier-Stokes equations, Jour. Math.Fluid Mechanics, 2 (2000), 16-98. doi: 10.1007/s000210050018. [5] A. Babin, Asymptotic Expansions at infinity of a strongly perturbed Poiseuille flow, Advances in Soviet Math., 10 (1992), 1-83. [6] A. Babin, The attractor of a Navier-Stokes system in an unbounded channel-like domain, J. Dynam. Differential Equations, 4 (1992), 555-584. doi: 10.1007/BF01048260. [7] A. Babin and M.Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243. doi: 10.1017/S0308210500031498. [8] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989; North Holland, Amsterdam, 1992. [9] M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688. doi: 10.1002/cpa.1011. [10] Y. Giga, S. Matsui and O. Sawada, Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity, J. Math. Fluid Mech., 3 (2001), 302-315. doi: 10.1007/PL00000973. [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. [12] P. Lemarie-Rieusset, Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, 431. Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674. [13] A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains - existence and comparison, Nonlinearity, 8 (1995), 743-768. [14] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations, Vol. IV, 103-200, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0. [15] J. Pennant and S. Zelik, Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $mathbb{R}^{3}$, Comm. Pure Appl. Anal., 12 (2013), 461-480. [16] S. Revina and V. Yudovich, $L^p$-estimates for the resolvent of the Stokes operator in an infinite cylinder, (Russian) Mat. Sb., 187 (1996), 97-118; translation in Sb. Math., 187 (1996), 881-902. doi: 10.1070/SM1996v187n06ABEH000139. [17] O. Sawada and Y. Taniuchi, A remark on $L^\infty$-solutions to the 2D Navier-Stokes equations, J. Math. Fluid Mech., 9 (2007), 533-542. doi: 10.1007/s00021-005-0212-4. [18] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam New York-Oxford, 1977. [19] R. Temam, Infnite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Series, Springer, New York-Berlin, 1988; 2nd ed., New York, 1997. [20] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978. [21] S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip, Glasg. Math. J., 49 (2007), 525-588. doi: 10.1017/S0017089507003849. [22] S. Zelik, Weak spatially nondecaying solutions of 3D Navier-Stokes equations in cylindrical domains, Instability in models connected with fluid flows. II, 255-327, Int. Math. Ser. (N. Y.), 7, Springer, New York, 2008. doi: 10.1007/978-0-387-75219-8_6. [23] S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math., 56 (2003), 584-637. doi: 10.1002/cpa.10068. [24] S. Zelik, Infinite energy solutions for damped Navier-Stokes equations in $\mathbb{R}^2$, Jour. Math. Fluid Mech., 15 (2013), 717-745. doi: 10.1007/s00021-013-0144-3.
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