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Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme
1. | Indiana University - Mathematics Department, Bloomington, IN 47405, United States |
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. |
show all references
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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