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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

Peter Anthony 1, and  Sergey Zelik 2,

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.
Keywords: Navier-Stokes equations, unbounded domains, infinite-energy solutions..
Mathematics Subject Classification: 35B40, 35B45.
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). RAIRO Model. Math. Anal. Numer., 23 (1989), 359-370.  Google Scholar

[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.  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, J. Math. Fluid Mech., 7 (2005), 51-67. doi: 10.1007/s00021-004-0131-9.  Google Scholar

[4]

H. Amann, On the strong solvability of the Navier-Stokes equations, Jour. Math.Fluid Mechanics, 2 (2000), 16-98. doi: 10.1007/s000210050018.  Google Scholar

[5]

A. Babin, Asymptotic Expansions at infinity of a strongly perturbed Poiseuille flow, Advances in Soviet Math., 10 (1992), 1-83.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989; North Holland, Amsterdam, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[13]

A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains - existence and comparison, Nonlinearity, 8 (1995), 743-768.  Google Scholar

[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.  Google Scholar

[15]

J. Pennant and S. Zelik, Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\R^3$, Comm. Pure Appl. Anal., 12 (2013), 461-480.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam New York-Oxford, 1977.  Google Scholar

[19]

R. Temam, Infnite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Series, Springer, New York-Berlin, 1988; 2nd ed., New York, 1997.  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, Glasg. Math. J., 49 (2007), 525-588. 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, 255-327, Int. Math. Ser. (N. Y.), 7, Springer, New York, 2008. 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, Comm. Pure Appl. Math., 56 (2003), 584-637. doi: 10.1002/cpa.10068.  Google Scholar

[24]

S. Zelik, Infinite energy solutions for damped Navier-Stokes equations in $\R^2$, Jour. Math. Fluid Mech., 15 (2013), 717-745. doi: 10.1007/s00021-013-0144-3.  Google Scholar

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). RAIRO Model. Math. Anal. Numer., 23 (1989), 359-370.  Google Scholar

[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.  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, J. Math. Fluid Mech., 7 (2005), 51-67. doi: 10.1007/s00021-004-0131-9.  Google Scholar

[4]

H. Amann, On the strong solvability of the Navier-Stokes equations, Jour. Math.Fluid Mechanics, 2 (2000), 16-98. doi: 10.1007/s000210050018.  Google Scholar

[5]

A. Babin, Asymptotic Expansions at infinity of a strongly perturbed Poiseuille flow, Advances in Soviet Math., 10 (1992), 1-83.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989; North Holland, Amsterdam, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[13]

A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains - existence and comparison, Nonlinearity, 8 (1995), 743-768.  Google Scholar

[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.  Google Scholar

[15]

J. Pennant and S. Zelik, Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\R^3$, Comm. Pure Appl. Anal., 12 (2013), 461-480.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam New York-Oxford, 1977.  Google Scholar

[19]

R. Temam, Infnite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Series, Springer, New York-Berlin, 1988; 2nd ed., New York, 1997.  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, Glasg. Math. J., 49 (2007), 525-588. 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, 255-327, Int. Math. Ser. (N. Y.), 7, Springer, New York, 2008. 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, Comm. Pure Appl. Math., 56 (2003), 584-637. doi: 10.1002/cpa.10068.  Google Scholar

[24]

S. Zelik, Infinite energy solutions for damped Navier-Stokes equations in $\R^2$, Jour. Math. Fluid Mech., 15 (2013), 717-745. doi: 10.1007/s00021-013-0144-3.  Google Scholar

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