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.

[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.

[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.

[4]

H. Amann, On the strong solvability of the Navier-Stokes equations,, \emph{Jour. Math.Fluid Mechanics}, 2 (2000), 16. doi: 10.1007/s000210050018.

[5]

A. Babin, Asymptotic Expansions at infinity of a strongly perturbed Poiseuille flow,, \emph{Advances in Soviet Math.}, 10 (1992), 1.

[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.

[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.

[8]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Nauka, (1989).

[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.

[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.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).

[12]

P. Lemarie-Rieusset, Recent developments in the Navier-Stokes problem,, Chapman $&$ Hall/CRC Research Notes in Mathematics, (2002). doi: 10.1201/9781420035674.

[13]

A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains - existence and comparison,, \emph{Nonlinearity}, 8 (1995), 743.

[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.

[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.

[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.

[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.

[18]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, North-Holland, (1977).

[19]

R. Temam, Infnite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematics Series, (1988).

[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,, \emph{Glasg. Math. J.}, 49 (2007), 525. 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, (2008), 255. doi: 10.1007/978-0-387-75219-8_6.

[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.

[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.

show all references

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.

[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.

[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.

[4]

H. Amann, On the strong solvability of the Navier-Stokes equations,, \emph{Jour. Math.Fluid Mechanics}, 2 (2000), 16. doi: 10.1007/s000210050018.

[5]

A. Babin, Asymptotic Expansions at infinity of a strongly perturbed Poiseuille flow,, \emph{Advances in Soviet Math.}, 10 (1992), 1.

[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.

[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.

[8]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Nauka, (1989).

[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.

[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.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).

[12]

P. Lemarie-Rieusset, Recent developments in the Navier-Stokes problem,, Chapman $&$ Hall/CRC Research Notes in Mathematics, (2002). doi: 10.1201/9781420035674.

[13]

A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains - existence and comparison,, \emph{Nonlinearity}, 8 (1995), 743.

[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.

[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.

[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.

[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.

[18]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, North-Holland, (1977).

[19]

R. Temam, Infnite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematics Series, (1988).

[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,, \emph{Glasg. Math. J.}, 49 (2007), 525. 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, (2008), 255. doi: 10.1007/978-0-387-75219-8_6.

[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.

[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.

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