American Institute of Mathematical Sciences

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March  2018, 7(1): 33-52. doi: 10.3934/eect.2018002

Continuous data assimilation algorithm for simplified Bardina model

Débora A. F. Albanez 1, and  Maicon J. Benvenutti 2,,

1. 

Departamento Acadêmico de Matemática, Universidade Tecnológica Federal do Paraná, Cornélio Procópio, PR, 86300-000, Brasil
2. 

Departamento de Matemática, Universidade Federal de Santa Catarina, Blumenau, SC, 89036-002, Brasil

* Corresponding author: m.benvenutti@ufsc.br.

Received  September 2016 Revised  September 2017 Published  January 2018

Fund Project: The second author was supported by FAPESP, Brazil.

We present a continuous data assimilation algorithm for three-dimensional viscous simplified Bardina turbulence model, based on the fact that dissipative dynamical systems possess finite degrees of freedom. We construct an approximating solution of simplified Barbina model through an interpolant operator which is obtained using observational data of the system. This interpolant is inserted to theoric model coupled to a relaxation parameter, and our main result provides conditions on the finite-dimensional spatial resolution of collected measurements sufficient to ensure that the approximating solution converges to the theoric solution of the model. Global well-posedness of approximating solutions and related results with degrees of freedom are also presented.

Keywords: Bardina equations, continuous data assimilation, degree of freedom, global solutions, interpolants.
Mathematics Subject Classification: 93C20, 35Q30, 76N10, 93C20, 37C50, 76B75, 34D06.
Citation: Débora A. F. Albanez, Maicon J. Benvenutti. Continuous data assimilation algorithm for simplified Bardina model. Evolution Equations and Control Theory, 2018, 7 (1) : 33-52. doi: 10.3934/eect.2018002
References:
[1]

D. A. F. Albanez, H. J. Nuzzenzveig Lopes and E. S. Titi, Continuous data assimilation for the three-dimensionl Navier-Stokes-$α$ model, Asymptotic Analysis, 97 (2016), 139-164.  doi: 10.3233/ASY-151351.

[2]

J. P. Aubin, Un téoréme de compacité, C.R. Acad. Sci. Paris Sér. I Math,, 256 (1963), 5042-5044. 

[3]

A. Azouani, E. Olson and E. S. Titi, Continuous data assimilation using General interpolant observables, Journal of Nonlinear Science, 24 (2014), 277-304.  doi: 10.1007/s00332-013-9189-y.

[4]

V. Barbu, Stabilization of Navier-Stokes Flows Springer-Verlag, New York, 2011.

[5]

J. Bardina, J. Ferziger and W. Reynolds, Improved subgrid scale models for large eddy simulation, American Institute of Aeronautics and Astronautics Paper, 80 (1980), 80-1357.  doi: 10.2514/6.1980-1357.

[6]

J. Bernard, Solutions globales variationnelles et classiqeus des fluides de grade deux, C.R. Acad. Sci. Paris, 327 (1998), 953-958.  doi: 10.1016/S0764-4442(99)80142-6.

[7]

Y. Cao, D. D. Holm and E. S. Titi, On the Clark-$α$ model of turbulence: Global regularity and long-time dynamics, Jornal of Turbulence 6 (2005), Paper 20, 11 pp.

[8]

Y. Cao, E. M. Lusanin and E. S. Titi, Global welll-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Communications in Mathematical Sciences, 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[9]

S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channel and pipes, Physics of fluids, 11 (1999), 2343-2353.  doi: 10.1063/1.870096.

[10]

S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Physica D: Nonlinear Phenomena, 133 (1999), 49-65.  doi: 10.1016/S0167-2789(99)00098-6.

[11]

S. Chen, D. D. Holm, L. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Physica D: Nonlinear Phenomena, 133 (1999), 66-83.  doi: 10.1016/S0167-2789(99)00099-8.

[12]

A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-$α$ model of turbulence, Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, 461 (2005), 629-649.  doi: 10.1098/rspa.2004.1373.

[13]

D. Cioranescu and V. Girault, Weak and classical solutions of a family of second grade fluids, International Journal of Non-Linear Mechanics, 32 (1997), 317-335.  doi: 10.1016/S0020-7462(96)00056-X.

[14]

P. Constantin and C. Foias, Navier-Stokes Equations Chicago Lectures in Mathematics, 1988.

[15]

R. Daley, Atmospheric Data Analysis Cambridge University Press, 1991. doi: 10. 4267/2042/51948.

[16]

H. Davies and R. Turner, Updating prediction models by dynamical relaxation: an examination of the technique, Quarterly Journal of the Royal Meteorological Society, 103 (1977), 225-245.  doi: 10.1002/qj.49710343602.

[17]

J. Domarkadzki, Navier-Stokes-alpha model: LES equations with nonlinear dispersion, Special Vol. Ercoftac Bull, 2001.

[18]

A. Farhat, E. Lunasin and E. S. Titi, A data assimilation algorithm: The paradigm of the 3D Leray-$α$ model of turbulence, preprint, arXiv:1702.01506v1.

[19]

A. Farhat, M. Jolly and E. S. Titi, Continuous data assimilation for the 2D Bénard convection through velocity measurements, Physica D: Nonlinear Phenomena, 303 (2015), 59-66.  doi: 10.1016/j.physd.2015.03.011.

[20]

A. Farhat, E. Lunasin and E. S. Titi, Abridged continuous data assimilation for the 2D Navier-Stokes equations utilizing measurements of only one component of the velocity field, Journal of Mathematical Fluid Mechanics, 18 (2016), 1-23.  doi: 10.1007/s00021-015-0225-6.

[21]

A. Farhat, E. Lunasin and E. S. Titi, Data assimilation algorithm for 3D Bénard convection in porous media emplying only temperature measurements, Journal of Mathematical Analysis and Applications, 438 (2016), 492-506.  doi: 10.1016/j.jmaa.2016.01.072.

[22]

C. Foias, D. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, Journal of Dynamics and Differential Equations, 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.

[23]

C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-$α$ model of fluid turbulence, Physica D: Nonlinear Phenomena, 152 (2001), 505-519.  doi: 10.1016/S0167-2789(01)00191-9.

[24]

C. Foias, O. Manley, R. Temam and Y. Treve, Asymptotic analysis of the Navier-Stokes equations, Physica D: Nonlinear Phenomena, 9 (1983), 157-188.  doi: 10.1016/0167-2789(83)90297-X.

[25]

C. Foias and R. Temam, Determination of the solutions of the Navier-Stokes equations by a set of nodal values, Mathematics of Computation, 43 (1984), 117-133.  doi: 10.1090/S0025-5718-1984-0744927-9.

[26]

C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34. 

[27]

C. Foias and R. Temam, Determining nodes, finite diference schemes and inertial manifolds, Nonlinearity, 4 (1991), 135-153.  doi: 10.1088/0951-7715/4/1/009.

[28]

A. Friedman, Partial Differential Equations Holt, Rinehart and Winston, Inc., New York-Montreal, Que. -London, 1969.

[29]

M. Gesho, E. Olson and E. S. Titi, A computational study of a data assimilation algorithm for the two-dimensional Navier-Stokes equations, Communications in Computational Physics, 19 (2016), 1094-1110. 

[30]

D. D. Holm and J. Marsden, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Physical Review Letters, 349 (1998), 4173-4177.  doi: 10.1103/PhysRevLett.80.4173.

[31]

D. A. Jones and E. S. Titi, Determining finite volume elements for the 2D Navier-Stokes equations, Physica D: Nonlinear Phenomena, 60 (1992), 165-174.  doi: 10.1016/0167-2789(92)90233-D.

[32]

D. A. Jones and E. S. Titi, Upper bounds on the number of determining modes, nodes and volume elements for the Navier-Stokes equations, Indiana University Mathematics Journal, 42 (1993), 875-887.  doi: 10.1512/iumj.1993.42.42039.

[33]

P. Korn, On degrees of freedom of certain conservative turbulence models for the Navier-Stokes equations, Journal of Mathematical Analysis and Applications, 378 (2011), 49-63.  doi: 10.1016/j.jmaa.2011.01.013.

[34]

W. Layton and R. Lewandowski, On a well-posed turbulence model, Discrete and Continuous Dynamical Systems -B, 6 (2006), 111-128. 

[35]

J. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic-$α$ models, Journal of Mathematical Physics 48 (2007), 065504, 28 pp.

[36]

J. L. Lions, Quelque Méthodes de Résolutions des Problémes aux Limites Non-Linéares Dunod, Paris, 1969.

[37]

E. Lunasin and E. Titi, Finite determining parameters feedback control for distributed nonlinear dissipative systems - a computational study, Evol. Equ. Control Theory, 6 (2017), 535-557, arXiv:1506.03709. doi: 10.3934/eect.2017027.

[38]

P. Markowich, E. S. Titi and S. Trablesi, Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model, Nonlinearity, 29 (2016), 1292-1328.  doi: 10.1088/0951-7715/29/4/1292.

[39]

I. Simmonds, H. Davies and R. Turner, Comments on the paper 'updating prediction models by dynamical relaxation: An examination of the technique' by H. C. Davies and R. E. Turner, I. Quart. J. R. Meteorological Society, 104 (1978), 527-532. 

[40]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis AMS Chelsea Publishing, Providence, RI, 2001.

[41]

Y. Yu and K. Li, Existence of solutions for the MHD-Leray-$α$ equations and their relations to the MHD equations, Journal of Mathematical Analysis and Applications, 329 (2007), 298-326.  doi: 10.1016/j.jmaa.2006.06.039.

show all references

References:
[1]

D. A. F. Albanez, H. J. Nuzzenzveig Lopes and E. S. Titi, Continuous data assimilation for the three-dimensionl Navier-Stokes-$α$ model, Asymptotic Analysis, 97 (2016), 139-164.  doi: 10.3233/ASY-151351.

[2]

J. P. Aubin, Un téoréme de compacité, C.R. Acad. Sci. Paris Sér. I Math,, 256 (1963), 5042-5044. 

[3]

A. Azouani, E. Olson and E. S. Titi, Continuous data assimilation using General interpolant observables, Journal of Nonlinear Science, 24 (2014), 277-304.  doi: 10.1007/s00332-013-9189-y.

[4]

V. Barbu, Stabilization of Navier-Stokes Flows Springer-Verlag, New York, 2011.

[5]

J. Bardina, J. Ferziger and W. Reynolds, Improved subgrid scale models for large eddy simulation, American Institute of Aeronautics and Astronautics Paper, 80 (1980), 80-1357.  doi: 10.2514/6.1980-1357.

[6]

J. Bernard, Solutions globales variationnelles et classiqeus des fluides de grade deux, C.R. Acad. Sci. Paris, 327 (1998), 953-958.  doi: 10.1016/S0764-4442(99)80142-6.

[7]

Y. Cao, D. D. Holm and E. S. Titi, On the Clark-$α$ model of turbulence: Global regularity and long-time dynamics, Jornal of Turbulence 6 (2005), Paper 20, 11 pp.

[8]

Y. Cao, E. M. Lusanin and E. S. Titi, Global welll-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Communications in Mathematical Sciences, 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[9]

S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channel and pipes, Physics of fluids, 11 (1999), 2343-2353.  doi: 10.1063/1.870096.

[10]

S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Physica D: Nonlinear Phenomena, 133 (1999), 49-65.  doi: 10.1016/S0167-2789(99)00098-6.

[11]

S. Chen, D. D. Holm, L. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Physica D: Nonlinear Phenomena, 133 (1999), 66-83.  doi: 10.1016/S0167-2789(99)00099-8.

[12]

A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-$α$ model of turbulence, Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, 461 (2005), 629-649.  doi: 10.1098/rspa.2004.1373.

[13]

D. Cioranescu and V. Girault, Weak and classical solutions of a family of second grade fluids, International Journal of Non-Linear Mechanics, 32 (1997), 317-335.  doi: 10.1016/S0020-7462(96)00056-X.

[14]

P. Constantin and C. Foias, Navier-Stokes Equations Chicago Lectures in Mathematics, 1988.

[15]

R. Daley, Atmospheric Data Analysis Cambridge University Press, 1991. doi: 10. 4267/2042/51948.

[16]

H. Davies and R. Turner, Updating prediction models by dynamical relaxation: an examination of the technique, Quarterly Journal of the Royal Meteorological Society, 103 (1977), 225-245.  doi: 10.1002/qj.49710343602.

[17]

J. Domarkadzki, Navier-Stokes-alpha model: LES equations with nonlinear dispersion, Special Vol. Ercoftac Bull, 2001.

[18]

A. Farhat, E. Lunasin and E. S. Titi, A data assimilation algorithm: The paradigm of the 3D Leray-$α$ model of turbulence, preprint, arXiv:1702.01506v1.

[19]

A. Farhat, M. Jolly and E. S. Titi, Continuous data assimilation for the 2D Bénard convection through velocity measurements, Physica D: Nonlinear Phenomena, 303 (2015), 59-66.  doi: 10.1016/j.physd.2015.03.011.

[20]

A. Farhat, E. Lunasin and E. S. Titi, Abridged continuous data assimilation for the 2D Navier-Stokes equations utilizing measurements of only one component of the velocity field, Journal of Mathematical Fluid Mechanics, 18 (2016), 1-23.  doi: 10.1007/s00021-015-0225-6.

[21]

A. Farhat, E. Lunasin and E. S. Titi, Data assimilation algorithm for 3D Bénard convection in porous media emplying only temperature measurements, Journal of Mathematical Analysis and Applications, 438 (2016), 492-506.  doi: 10.1016/j.jmaa.2016.01.072.

[22]

C. Foias, D. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, Journal of Dynamics and Differential Equations, 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.

[23]

C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-$α$ model of fluid turbulence, Physica D: Nonlinear Phenomena, 152 (2001), 505-519.  doi: 10.1016/S0167-2789(01)00191-9.

[24]

C. Foias, O. Manley, R. Temam and Y. Treve, Asymptotic analysis of the Navier-Stokes equations, Physica D: Nonlinear Phenomena, 9 (1983), 157-188.  doi: 10.1016/0167-2789(83)90297-X.

[25]

C. Foias and R. Temam, Determination of the solutions of the Navier-Stokes equations by a set of nodal values, Mathematics of Computation, 43 (1984), 117-133.  doi: 10.1090/S0025-5718-1984-0744927-9.

[26]

C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34. 

[27]

C. Foias and R. Temam, Determining nodes, finite diference schemes and inertial manifolds, Nonlinearity, 4 (1991), 135-153.  doi: 10.1088/0951-7715/4/1/009.

[28]

A. Friedman, Partial Differential Equations Holt, Rinehart and Winston, Inc., New York-Montreal, Que. -London, 1969.

[29]

M. Gesho, E. Olson and E. S. Titi, A computational study of a data assimilation algorithm for the two-dimensional Navier-Stokes equations, Communications in Computational Physics, 19 (2016), 1094-1110. 

[30]

D. D. Holm and J. Marsden, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Physical Review Letters, 349 (1998), 4173-4177.  doi: 10.1103/PhysRevLett.80.4173.

[31]

D. A. Jones and E. S. Titi, Determining finite volume elements for the 2D Navier-Stokes equations, Physica D: Nonlinear Phenomena, 60 (1992), 165-174.  doi: 10.1016/0167-2789(92)90233-D.

[32]

D. A. Jones and E. S. Titi, Upper bounds on the number of determining modes, nodes and volume elements for the Navier-Stokes equations, Indiana University Mathematics Journal, 42 (1993), 875-887.  doi: 10.1512/iumj.1993.42.42039.

[33]

P. Korn, On degrees of freedom of certain conservative turbulence models for the Navier-Stokes equations, Journal of Mathematical Analysis and Applications, 378 (2011), 49-63.  doi: 10.1016/j.jmaa.2011.01.013.

[34]

W. Layton and R. Lewandowski, On a well-posed turbulence model, Discrete and Continuous Dynamical Systems -B, 6 (2006), 111-128. 

[35]

J. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic-$α$ models, Journal of Mathematical Physics 48 (2007), 065504, 28 pp.

[36]

J. L. Lions, Quelque Méthodes de Résolutions des Problémes aux Limites Non-Linéares Dunod, Paris, 1969.

[37]

E. Lunasin and E. Titi, Finite determining parameters feedback control for distributed nonlinear dissipative systems - a computational study, Evol. Equ. Control Theory, 6 (2017), 535-557, arXiv:1506.03709. doi: 10.3934/eect.2017027.

[38]

P. Markowich, E. S. Titi and S. Trablesi, Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model, Nonlinearity, 29 (2016), 1292-1328.  doi: 10.1088/0951-7715/29/4/1292.

[39]

I. Simmonds, H. Davies and R. Turner, Comments on the paper 'updating prediction models by dynamical relaxation: An examination of the technique' by H. C. Davies and R. E. Turner, I. Quart. J. R. Meteorological Society, 104 (1978), 527-532. 

[40]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis AMS Chelsea Publishing, Providence, RI, 2001.

[41]

Y. Yu and K. Li, Existence of solutions for the MHD-Leray-$α$ equations and their relations to the MHD equations, Journal of Mathematical Analysis and Applications, 329 (2007), 298-326.  doi: 10.1016/j.jmaa.2006.06.039.

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