# American Institute of Mathematical Sciences

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

 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.
Citation: Michele Coti Zelati. Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2829-2838. doi: 10.3934/cpaa.2013.12.2829
##### References:
 [1] J.-P. Aubin and H. Frankowska, "Set-valued Analysis," Birkhäuser, Boston, 1990. [2] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland Publishing Co., Amsterdam, 1992. [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. [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. [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. [6] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," American Mathematical Society, Providence, 2002. [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. [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. [9] C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes Equations and Turbulence," Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754. [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. [11] J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, 1988. [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. [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. [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. [15] M. Marion and R. Temam, Navier-Stokes equations: theory and approximation, in "Handbook of Numerical Analysis, Vol. VI," North-Holland, (1998), 503-688. [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. [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. [18] J. C. Robinson, "Infinite-dimensional Dynamical Systems," Cambridge University Press, Cambridge, 2001. [19] H. Sohr, "The Navier-Stokes Equations," Birkhäuser Verlag, Basel, 2001. [20] R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970050. [21] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, 2001. [22] R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1997. [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. [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. [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. [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. [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.

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##### References:
 [1] J.-P. Aubin and H. Frankowska, "Set-valued Analysis," Birkhäuser, Boston, 1990. [2] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland Publishing Co., Amsterdam, 1992. [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. [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. [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. [6] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," American Mathematical Society, Providence, 2002. [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. [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. [9] C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes Equations and Turbulence," Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754. [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. [11] J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, 1988. [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. [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. [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. [15] M. Marion and R. Temam, Navier-Stokes equations: theory and approximation, in "Handbook of Numerical Analysis, Vol. VI," North-Holland, (1998), 503-688. [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. [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. [18] J. C. Robinson, "Infinite-dimensional Dynamical Systems," Cambridge University Press, Cambridge, 2001. [19] H. Sohr, "The Navier-Stokes Equations," Birkhäuser Verlag, Basel, 2001. [20] R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970050. [21] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, 2001. [22] R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1997. [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. [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. [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. [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. [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.
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