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November  2013, 12(6): 2829-2838. doi: 10.3934/cpaa.2013.12.2829

Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme

Michele Coti Zelati 1,

1. 

Indiana University - Mathematics Department, Bloomington, IN 47405, United States

Received  November 2012 Revised  March 2013 Published  May 2013

In this short note, we exploit the tools of multivalued dynamical systems to prove that the stationary statistical properties of the fully implicit Euler scheme converge, as the time-step parameter vanishes, to the stationary statistical properties of the two-dimensional Navier-Stokes equations.
Keywords: Multivalued dynamical systems, Navier-Stokes equations, invariant measures, implicit Euler scheme., time discretization.
Mathematics Subject Classification: Primary: 37M25, 37L40; Secondary: 47H04, 47H20, 76D0.
Citation: Michele Coti Zelati. Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2829-2838. doi: 10.3934/cpaa.2013.12.2829
References:
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J.-P. Aubin and H. Frankowska, "Set-valued Analysis," Birkhäuser, Boston, 1990.  Google Scholar

[2]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

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T. Caraballo, P. Marín-Rubio and J. C. Robinson, A comparison between two theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Anal., 11 (2003), 297-322. doi: 10.1023/A:1024422619616.  Google Scholar

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M. D. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: abstract results and applications, Comm. Math. Phys, 316 (2012), 723-761. doi: 10.1007/s00220-012-1515-y.  Google Scholar

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V. V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractors, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088. doi: 10.3934/dcds.2012.32.2079.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," American Mathematical Society, Providence, 2002.  Google Scholar

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M. Coti Zelati, On the theory of global attractors and lyapunov functionals, Set-Valued Var. Anal., 21 (2013), 127-149. doi: 10.1007/s11228-012-0215-2.  Google Scholar

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M. Coti Zelati and F. Tone, Multivalued attractors and their approximation: applications to the Navier-Stokes equations, Numer. Math., 122 (2012), 421-441. doi: 10.1007/s00211-012-0463-y.  Google Scholar

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C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes Equations and Turbulence," Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.  Google Scholar

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C. B. Gentile and J. Simsen, On attractors for multivalued semigroups defined by generalized semiflows, Set-Valued Anal., 16 (2008), 105-124. doi: 10.1007/s11228-006-0037-1.  Google Scholar

[11]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, 1988.  Google Scholar

[12]

N. Ju, On the global stability of a temporal discretization scheme for the Navier-Stokes equations, IMA J. Numer. Anal., 22 (2002), 577-597. doi: 10.1093/imanum/22.4.577.  Google Scholar

[13]

A. V. Kapustian and J. Valero, Attractors of multivalued semiflows generated by differential inclusions and their approximations, Abstr. Appl. Anal., 5 (2000), 33-46. doi: 10.1155/S1085337500000191.  Google Scholar

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G. Łukaszewicz, J. Real and J. C. Robinson, Invariant measures for dissipative systems and generalised Banach limits, J. Dynam. Differential Equations, 23 (2011), 225-250. doi: 10.1007/s10884-011-9213-6.  Google Scholar

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M. Marion and R. Temam, Navier-Stokes equations: theory and approximation, in "Handbook of Numerical Analysis, Vol. VI," North-Holland, (1998), 503-688.  Google Scholar

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V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399.  Google Scholar

[17]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486. doi: 10.3934/cpaa.2007.6.481.  Google Scholar

[18]

J. C. Robinson, "Infinite-dimensional Dynamical Systems," Cambridge University Press, Cambridge, 2001.  Google Scholar

[19]

H. Sohr, "The Navier-Stokes Equations," Birkhäuser Verlag, Basel, 2001.  Google Scholar

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R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970050.  Google Scholar

[21]

R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, 2001.  Google Scholar

[22]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1997.  Google Scholar

[23]

F. Tone and X. Wang, Approximation of the stationary statistical properties of the dynamical system generated by the two-dimensional Rayleigh-Bénard convection problem, Anal. Appl. (Singap.), 9 (2011), 421-446. doi: 10.1142/S0219530511001935.  Google Scholar

[24]

F. Tone and D. Wirosoetisno, On the long-time stability of the implicit Euler scheme for the two-dimensional Navier-Stokes equations, SIAM J. Numer. Anal., 44 (2006), 29-40. doi: 10.1137/040618527.  Google Scholar

[25]

X. Wang, Upper semi-continuity of stationary statistical properties of dissipative systems, Discrete Contin. Dyn. Syst., 23 (2009), 521-540. doi: 10.3934/dcds.2009.23.521.  Google Scholar

[26]

X. Wang, Approximation of stationary statistical properties of dissipative dynamical systems: time discretization, Math. Comp., 79 (2010), 259-280. doi: 10.1090/S0025-5718-09-02256-X.  Google Scholar

[27]

X. Wang, An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations, Numer. Math., 121 (2012), 753-779. doi: 10.1007/s00211-012-0450-3.  Google Scholar

show all references

References:
[1]

J.-P. Aubin and H. Frankowska, "Set-valued Analysis," Birkhäuser, Boston, 1990.  Google Scholar

[2]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[3]

T. Caraballo, P. Marín-Rubio and J. C. Robinson, A comparison between two theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Anal., 11 (2003), 297-322. doi: 10.1023/A:1024422619616.  Google Scholar

[4]

M. D. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: abstract results and applications, Comm. Math. Phys, 316 (2012), 723-761. doi: 10.1007/s00220-012-1515-y.  Google Scholar

[5]

V. V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractors, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088. doi: 10.3934/dcds.2012.32.2079.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," American Mathematical Society, Providence, 2002.  Google Scholar

[7]

M. Coti Zelati, On the theory of global attractors and lyapunov functionals, Set-Valued Var. Anal., 21 (2013), 127-149. doi: 10.1007/s11228-012-0215-2.  Google Scholar

[8]

M. Coti Zelati and F. Tone, Multivalued attractors and their approximation: applications to the Navier-Stokes equations, Numer. Math., 122 (2012), 421-441. doi: 10.1007/s00211-012-0463-y.  Google Scholar

[9]

C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes Equations and Turbulence," Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.  Google Scholar

[10]

C. B. Gentile and J. Simsen, On attractors for multivalued semigroups defined by generalized semiflows, Set-Valued Anal., 16 (2008), 105-124. doi: 10.1007/s11228-006-0037-1.  Google Scholar

[11]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, 1988.  Google Scholar

[12]

N. Ju, On the global stability of a temporal discretization scheme for the Navier-Stokes equations, IMA J. Numer. Anal., 22 (2002), 577-597. doi: 10.1093/imanum/22.4.577.  Google Scholar

[13]

A. V. Kapustian and J. Valero, Attractors of multivalued semiflows generated by differential inclusions and their approximations, Abstr. Appl. Anal., 5 (2000), 33-46. doi: 10.1155/S1085337500000191.  Google Scholar

[14]

G. Łukaszewicz, J. Real and J. C. Robinson, Invariant measures for dissipative systems and generalised Banach limits, J. Dynam. Differential Equations, 23 (2011), 225-250. doi: 10.1007/s10884-011-9213-6.  Google Scholar

[15]

M. Marion and R. Temam, Navier-Stokes equations: theory and approximation, in "Handbook of Numerical Analysis, Vol. VI," North-Holland, (1998), 503-688.  Google Scholar

[16]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399.  Google Scholar

[17]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486. doi: 10.3934/cpaa.2007.6.481.  Google Scholar

[18]

J. C. Robinson, "Infinite-dimensional Dynamical Systems," Cambridge University Press, Cambridge, 2001.  Google Scholar

[19]

H. Sohr, "The Navier-Stokes Equations," Birkhäuser Verlag, Basel, 2001.  Google Scholar

[20]

R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970050.  Google Scholar

[21]

R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, 2001.  Google Scholar

[22]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1997.  Google Scholar

[23]

F. Tone and X. Wang, Approximation of the stationary statistical properties of the dynamical system generated by the two-dimensional Rayleigh-Bénard convection problem, Anal. Appl. (Singap.), 9 (2011), 421-446. doi: 10.1142/S0219530511001935.  Google Scholar

[24]

F. Tone and D. Wirosoetisno, On the long-time stability of the implicit Euler scheme for the two-dimensional Navier-Stokes equations, SIAM J. Numer. Anal., 44 (2006), 29-40. doi: 10.1137/040618527.  Google Scholar

[25]

X. Wang, Upper semi-continuity of stationary statistical properties of dissipative systems, Discrete Contin. Dyn. Syst., 23 (2009), 521-540. doi: 10.3934/dcds.2009.23.521.  Google Scholar

[26]

X. Wang, Approximation of stationary statistical properties of dissipative dynamical systems: time discretization, Math. Comp., 79 (2010), 259-280. doi: 10.1090/S0025-5718-09-02256-X.  Google Scholar

[27]

X. Wang, An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations, Numer. Math., 121 (2012), 753-779. doi: 10.1007/s00211-012-0450-3.  Google Scholar

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