American Institute of Mathematical Sciences

Advanced
March  2010, 28(1): 375-403. doi: 10.3934/dcds.2010.28.375

Invariant measures for the $3$D Navier-Stokes-Voigt equations and their Navier-Stokes limit

Fabio Ramos 1, and  Edriss S. Titi 1,

1. 

Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel, Israel

Received  October 2009 Revised  February 2010 Published  April 2010

The Navier-Stokes-Voigt model of viscoelastic incompressible fluid has been recently proposed as a regularization of the three-dimensional Navier-Stokes equations for the purpose of direct numerical simulations. Besides the kinematic viscosity parameter, $\nu>0$, this model possesses a regularizing parameter, $\alpha> 0$, a given length scale parameter, so that $\frac{\alpha^2}{\nu}$ is the relaxation time of the viscoelastic fluid. In this work, we derive several statistical properties of the invariant measures associated with the solutions of the three-dimensional Navier-Stokes-Voigt equations. Moreover, we prove that, for fixed viscosity , $\nu>0$, as the regularizing parameter $\alpha$ tends to zero, there exists a subsequence of probability invariant measures converging, in a suitable sense, to a strong stationary statistical solution of the three-dimensional Navier-Stokes equations, which is a regularized version of the notion of stationary statistical solutions - a generalization of the concept of invariant measure introduced and investigated by Foias. This fact supports earlier numerical observations, and provides an additional evidence that, for small values of the regularization parameter $\alpha$, the Navier-Stokes-Voigt model can indeed be considered as a model to study the statistical properties of the three-dimensional Navier-Stokes equations and turbulent flows via direct numerical simulations.
Keywords: Navier-Stokes-Voigt, Navier-Stokes-Voight, inviscid regularization, viscoelastic flows., Statistical solutions, turbulence.
Mathematics Subject Classification: 76D06, 76D05, 76F20, 76F55, 76A1.
Citation: Fabio Ramos, Edriss S. Titi. Invariant measures for the $3$D Navier-Stokes-Voigt equations and their Navier-Stokes limit. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 375-403. doi: 10.3934/dcds.2010.28.375
[1]

Luigi C. Berselli, Tae-Yeon Kim, Leo G. Rebholz. Analysis of a reduced-order approximate deconvolution model and its interpretation as a Navier-Stokes-Voigt regularization. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1027-1050. doi: 10.3934/dcdsb.2016.21.1027
[2]

Gaocheng Yue, Chengkui Zhong. Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 985-1002. doi: 10.3934/dcdsb.2011.16.985
[3]

Varga K. Kalantarov, Edriss S. Titi. Global stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1325-1345. doi: 10.3934/dcdsb.2018153
[4]

Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations and Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039
[5]

Xinguang Yang, Baowei Feng, Thales Maier de Souza, Taige Wang. Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equation in Lipschitz domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 363-386. doi: 10.3934/dcdsb.2018084
[6]

Vu Manh Toi. Stability and stabilization for the three-dimensional Navier-Stokes-Voigt equations with unbounded variable delay. Evolution Equations and Control Theory, 2021, 10 (4) : 1007-1023. doi: 10.3934/eect.2020099
[7]

Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761
[8]

Adam Larios, Yuan Pei. Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data. Evolution Equations and Control Theory, 2020, 9 (3) : 733-751. doi: 10.3934/eect.2020031
[9]

Yat Tin Chow, Ali Pakzad. On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021270
[10]

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 and Applied Analysis, 2009, 8 (3) : 785-802. doi: 10.3934/cpaa.2009.8.785
[11]

Anne Bronzi, Ricardo Rosa. On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 19-49. doi: 10.3934/dcds.2014.34.19
[12]

Grzegorz Łukaszewicz. Pullback attractors and statistical solutions for 2-D Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 643-659. doi: 10.3934/dcdsb.2008.9.643
[13]

Bo You. Trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021251
[14]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673
[15]

Wenjing Song, Ganshan Yang. The regularization of solution for the coupled Navier-Stokes and Maxwell equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2113-2127. doi: 10.3934/dcdss.2016087
[16]

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

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747
[18]

Tian Ma, Shouhong Wang. Asymptotic structure for solutions of the Navier--Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 189-204. doi: 10.3934/dcds.2004.11.189
[19]

Eric Blayo, Antoine Rousseau. About interface conditions for coupling hydrostatic and nonhydrostatic Navier-Stokes flows. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1565-1574. doi: 10.3934/dcdss.2016063
[20]

Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (119)
  • HTML views (0)
  • Cited by (20)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy