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

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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

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

Received November 2012 Revised March 2013 Published May 2013

Abstract
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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:

[1]	J.-P. Aubin and H. Frankowska, "Set-valued Analysis,", Birkh\, (1990). Google Scholar
[2]	A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland Publishing Co., (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. 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. 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. 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, (2002). Google Scholar
[7]	M. Coti Zelati, On the theory of global attractors and lyapunov functionals,, Set-Valued Var. Anal., 21 (2013), 127. 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. 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, (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. doi: 10.1007/s11228-006-0037-1. Google Scholar
[11]	J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", American Mathematical Society, (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. 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. 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. doi: 10.1007/s10884-011-9213-6. Google Scholar
[15]	M. Marion and R. Temam, Navier-Stokes equations: theory and approximation,, in, (1998), 503. Google Scholar
[16]	V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions,, Set-Valued Anal., 6 (1998), 83. 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. doi: 10.3934/cpaa.2007.6.481. Google Scholar
[18]	J. C. Robinson, "Infinite-dimensional Dynamical Systems,", Cambridge University Press, (2001). Google Scholar
[19]	H. Sohr, "The Navier-Stokes Equations,", Birkh\, (2001). Google Scholar
[20]	R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", SIAM, (1995). doi: 10.1137/1.9781611970050. Google Scholar
[21]	R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", AMS Chelsea Publishing, (2001). Google Scholar
[22]	R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,", Springer-Verlag, (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. 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. 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. 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. 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. doi: 10.1007/s00211-012-0450-3. Google Scholar

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References:

[1]	J.-P. Aubin and H. Frankowska, "Set-valued Analysis,", Birkh\, (1990). Google Scholar
[2]	A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland Publishing Co., (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. 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. 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. 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, (2002). Google Scholar
[7]	M. Coti Zelati, On the theory of global attractors and lyapunov functionals,, Set-Valued Var. Anal., 21 (2013), 127. 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. 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, (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. doi: 10.1007/s11228-006-0037-1. Google Scholar
[11]	J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", American Mathematical Society, (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. 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. 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. doi: 10.1007/s10884-011-9213-6. Google Scholar
[15]	M. Marion and R. Temam, Navier-Stokes equations: theory and approximation,, in, (1998), 503. Google Scholar
[16]	V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions,, Set-Valued Anal., 6 (1998), 83. 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. doi: 10.3934/cpaa.2007.6.481. Google Scholar
[18]	J. C. Robinson, "Infinite-dimensional Dynamical Systems,", Cambridge University Press, (2001). Google Scholar
[19]	H. Sohr, "The Navier-Stokes Equations,", Birkh\, (2001). Google Scholar
[20]	R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", SIAM, (1995). doi: 10.1137/1.9781611970050. Google Scholar
[21]	R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", AMS Chelsea Publishing, (2001). Google Scholar
[22]	R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,", Springer-Verlag, (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. 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. 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. 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. 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. doi: 10.1007/s00211-012-0450-3. Google Scholar

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