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 & 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

show all references

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

[1]

Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173

[2]

Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761

[3]

Fabio Ramos, Edriss S. Titi. Invariant measures for the $3$D Navier-Stokes-Voigt equations and their Navier-Stokes limit. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 375-403. doi: 10.3934/dcds.2010.28.375

[4]

P.E. Kloeden, Pedro Marín-Rubio, José Real. Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 785-802. doi: 10.3934/cpaa.2009.8.785

[5]

Yi Zhou, Zhen Lei. Logarithmically improved criteria for Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2715-2719. doi: 10.3934/cpaa.2013.12.2715

[6]

Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557

[7]

V. V. Chepyzhov, A. A. Ilyin. On the fractal dimension of invariant sets: Applications to Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 117-135. doi: 10.3934/dcds.2004.10.117

[8]

Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353

[9]

Takeshi Taniguchi. The exponential behavior of Navier-Stokes equations with time delay external force. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 997-1018. doi: 10.3934/dcds.2005.12.997

[10]

Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073

[11]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[12]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

[13]

Dongho Chae, Kyungkeun Kang, Jihoon Lee. Notes on the asymptotically self-similar singularities in the Euler and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1181-1193. doi: 10.3934/dcds.2009.25.1181

[14]

Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285

[15]

Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349

[16]

Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433

[17]

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747

[18]

C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403

[19]

Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319

[20]

Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]