August  2022, 42(8): 3661-3707. doi: 10.3934/dcds.2022028

On the long-time behavior for a damped Navier-Stokes-Bardina model

1. 

Escuela de Ciencias Físicas y Matemáticas, Universidad de Las Américas, Vía a Nayón, C.P.170124, Quito, Ecuador

2. 

Departamento de Matemáticas, Escuela Politécnica Nacional, Ladrón de Guevera E11-253, Quito, Ecuador

*Corresponding author: Oscar Jarrín

Received  July 2021 Revised  January 2022 Published  August 2022 Early access  March 2022

We consider a damped Navier-Stokes-Bardina model posed on the whole three-dimensional space. These equations write down as the well-know Navier-Stokes equations with an additional nonlocal operator in the nonlinear transport term, and moreover, with an additional damping term depending on a parameter $ \beta>0 $. First, we study the existence and uniqueness of global in time weak solutions in the energy space. Thereafter, our main objective is to describe the long time behavior of these solutions. For this, we use some tools in the theory of dynamical systems to prove the existence of a global attractor, which is compact subset in the energy space attracting all the weak solutions when the time goes to infinity. Moreover, we derive an upper bound for the fractal dimension of the global attractor associated to these equations.

Finally, we find a range of values for the damping parameter $ \beta>0 $, for which we are able to give an acute description of the internal structure of the global attractor. More precisely, we prove that in some cases the global attractor only contains the stationary (time-independent) solution of the damped Navier-Stokes-Bardina equations.

Citation: Oscar Jarrín, Manuel Fernando Cortez. On the long-time behavior for a damped Navier-Stokes-Bardina model. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3661-3707. doi: 10.3934/dcds.2022028
References:
[1]

N. A. Adams and S. Stolz, Deconvolution methods for subgrid-scale approximation in large eddy simulation, Modern Simulation Strategies for Turbulent Flow, R.T. Edwards, Academic Press, (2001).

[2]

A. V. Babin and M. I. Vishik, Attractors of evolution partial differential equations and estimates of their dimension, Uspekhi Mat. Nauk, 38 (2001) 133–187.

[3]

A. V. Babin and M. I. Vishik, Atracttors for Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[4]

J. Bardina, J. Coakley and P. Huang, Turbulence Modeling Validation, Testing, and Development, NASA Technical Memorandum, 1997.

[5]

J. BardinaJ. Ferziger and W. Reynolds, Improved subgrid scale models for large eddy simulation, AIAA, 80 (1980), 1300-1357.  doi: 10.2514/6.1980-1357.

[6]

Y. CaoE. M. Lunasin and Edriss S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci, 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[7]

D. ChamorroO. Jarrín and P.-G. Lemarié-Rieusset, On the Kolmogorov dissipation law in a damped Navier-Stokes equation, Journal of Dynamics and Differential Equation, 33 (2021), 1109-1134.  doi: 10.1007/s10884-020-09851-6.

[8]

V. V. Chepyzhov and A. A. Ilyin, On the fractal dimension of invariant sets: Applications to Navier-Stokes equations, Discrete Contin. Dyn. Syst, 10 (2004), 117-135.  doi: 10.3934/dcds.2004.10.117.

[9]

F. K. Chow, S. F. J. De Wekker and B. J. Synder, Mountain Weather Research and Forecasting: Recent Progress and Current Challenges, Springer, Berlin, 2013. doi: 10.1007/978-94-007-4098-3.

[10]

P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for the 2D Navier-Stokes equations, Comm. Pure Appl. Math, 38 (1985), 1-27.  doi: 10.1002/cpa.3160380102.

[11]

P. Constantin and F. Ramos, Inviscid limit for damped and driven incompressible Navier-Stokes equations in $\mathbb{R}^2$, Comm. Math. Phys., 275 (2007), 529-551.  doi: 10.1007/s00220-007-0310-7.

[12]

S. De BriéveF. Genoud and S. Rota Nodari, Orbital stability: Analysis meets geometry, Nonlinear Optical and Atomic Systems, Lectures Notes in Mathematics, 2146 (2015), 147-273.  doi: 10.1007/978-3-319-19015-0_3.

[13]

L. Grafakos, Modern Fourier Analysis, Second Edition. Springer Series 250, 2008. doi: 10.1007/978-0-387-09434-2.

[14]

A. A. IlyinE. M. Lunasin and E. S. Titi, A modified-Leray-$\alpha$ subgrid scale model of turbulence, Nonlinearity, 19 (2006), 879-897.  doi: 10.1088/0951-7715/19/4/006.

[15]

A. IlynK. Patni and S. Zelik, Upper bounds for the attractor dimension of damped Navier-Stokes equations in ${\mathbb{R}}^2$, Discrete and Continuous Dynamical Systems, 36 (2016), 2085-2102.  doi: 10.3934/dcds.2016.36.2085.

[16]

O. Jarrín., Deterministic Descriptions of Turbulence in the Navier-Stokes Equations, Ph.D. thesis, Université Paris-Saclay, Evry 2018.

[17]

W. Layton and R. Lewandowski, A simple and stable scale-similarity model for large eddy simulation: Energy balance and existence of weak solutions, Appl. Math. Lett, 16 (2003), 1205-1209.  doi: 10.1016/S0893-9659(03)90118-2.

[18]

W. Layton and R. Lewandowski, On a well-posed turbulence model, Dicrete and Continuous Dyn. Sys. B, 6 (2006), 111-128.  doi: 10.3934/dcdsb.2006.6.111.

[19] P. G. Lemarié-Rieusset, The Navier–Stokes Problem in the 21st Century, CRC Press, Chapman & Hall Book, 2016.  doi: 10.1201/b19556.
[20]

R. Lewandowski and L. C. Berselli, On the Bardina's model in the whole space, Journal of Mathematical Fluid Mechanics, 20 (2018), 1335-1351.  doi: 10.1007/s00021-018-0369-2.

[21]

E. H. Lieb, Lieb-Thirring Inequalities, preprint, https://arXiv.org/pdf/math-ph/0003039.pdf arXiv: math-ph/0003039

[22]

H. Liua and H. Gao, Decay of solutions for the 3D Navier-Stokes equations with damping, Applied Mathematics Letters, 68 (2017), 48-54.  doi: 10.1016/j.aml.2016.11.013.

[23]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979. doi: 10.1007/978-1-4684-0071-7.

[24]

G. Raugel, Global Attractors in Partial Differential Equations, Lectures notes, CNRS et Université Paris-Sud, Analyse Numérique et EDP, UMR 8628, 2006.

[25]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2$^nd$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[26]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.

show all references

References:
[1]

N. A. Adams and S. Stolz, Deconvolution methods for subgrid-scale approximation in large eddy simulation, Modern Simulation Strategies for Turbulent Flow, R.T. Edwards, Academic Press, (2001).

[2]

A. V. Babin and M. I. Vishik, Attractors of evolution partial differential equations and estimates of their dimension, Uspekhi Mat. Nauk, 38 (2001) 133–187.

[3]

A. V. Babin and M. I. Vishik, Atracttors for Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[4]

J. Bardina, J. Coakley and P. Huang, Turbulence Modeling Validation, Testing, and Development, NASA Technical Memorandum, 1997.

[5]

J. BardinaJ. Ferziger and W. Reynolds, Improved subgrid scale models for large eddy simulation, AIAA, 80 (1980), 1300-1357.  doi: 10.2514/6.1980-1357.

[6]

Y. CaoE. M. Lunasin and Edriss S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci, 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[7]

D. ChamorroO. Jarrín and P.-G. Lemarié-Rieusset, On the Kolmogorov dissipation law in a damped Navier-Stokes equation, Journal of Dynamics and Differential Equation, 33 (2021), 1109-1134.  doi: 10.1007/s10884-020-09851-6.

[8]

V. V. Chepyzhov and A. A. Ilyin, On the fractal dimension of invariant sets: Applications to Navier-Stokes equations, Discrete Contin. Dyn. Syst, 10 (2004), 117-135.  doi: 10.3934/dcds.2004.10.117.

[9]

F. K. Chow, S. F. J. De Wekker and B. J. Synder, Mountain Weather Research and Forecasting: Recent Progress and Current Challenges, Springer, Berlin, 2013. doi: 10.1007/978-94-007-4098-3.

[10]

P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for the 2D Navier-Stokes equations, Comm. Pure Appl. Math, 38 (1985), 1-27.  doi: 10.1002/cpa.3160380102.

[11]

P. Constantin and F. Ramos, Inviscid limit for damped and driven incompressible Navier-Stokes equations in $\mathbb{R}^2$, Comm. Math. Phys., 275 (2007), 529-551.  doi: 10.1007/s00220-007-0310-7.

[12]

S. De BriéveF. Genoud and S. Rota Nodari, Orbital stability: Analysis meets geometry, Nonlinear Optical and Atomic Systems, Lectures Notes in Mathematics, 2146 (2015), 147-273.  doi: 10.1007/978-3-319-19015-0_3.

[13]

L. Grafakos, Modern Fourier Analysis, Second Edition. Springer Series 250, 2008. doi: 10.1007/978-0-387-09434-2.

[14]

A. A. IlyinE. M. Lunasin and E. S. Titi, A modified-Leray-$\alpha$ subgrid scale model of turbulence, Nonlinearity, 19 (2006), 879-897.  doi: 10.1088/0951-7715/19/4/006.

[15]

A. IlynK. Patni and S. Zelik, Upper bounds for the attractor dimension of damped Navier-Stokes equations in ${\mathbb{R}}^2$, Discrete and Continuous Dynamical Systems, 36 (2016), 2085-2102.  doi: 10.3934/dcds.2016.36.2085.

[16]

O. Jarrín., Deterministic Descriptions of Turbulence in the Navier-Stokes Equations, Ph.D. thesis, Université Paris-Saclay, Evry 2018.

[17]

W. Layton and R. Lewandowski, A simple and stable scale-similarity model for large eddy simulation: Energy balance and existence of weak solutions, Appl. Math. Lett, 16 (2003), 1205-1209.  doi: 10.1016/S0893-9659(03)90118-2.

[18]

W. Layton and R. Lewandowski, On a well-posed turbulence model, Dicrete and Continuous Dyn. Sys. B, 6 (2006), 111-128.  doi: 10.3934/dcdsb.2006.6.111.

[19] P. G. Lemarié-Rieusset, The Navier–Stokes Problem in the 21st Century, CRC Press, Chapman & Hall Book, 2016.  doi: 10.1201/b19556.
[20]

R. Lewandowski and L. C. Berselli, On the Bardina's model in the whole space, Journal of Mathematical Fluid Mechanics, 20 (2018), 1335-1351.  doi: 10.1007/s00021-018-0369-2.

[21]

E. H. Lieb, Lieb-Thirring Inequalities, preprint, https://arXiv.org/pdf/math-ph/0003039.pdf arXiv: math-ph/0003039

[22]

H. Liua and H. Gao, Decay of solutions for the 3D Navier-Stokes equations with damping, Applied Mathematics Letters, 68 (2017), 48-54.  doi: 10.1016/j.aml.2016.11.013.

[23]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979. doi: 10.1007/978-1-4684-0071-7.

[24]

G. Raugel, Global Attractors in Partial Differential Equations, Lectures notes, CNRS et Université Paris-Sud, Analyse Numérique et EDP, UMR 8628, 2006.

[25]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2$^nd$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[26]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.

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