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September  2017, 16(5): 1861-1881. doi: 10.3934/cpaa.2017090

Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region

Luca Bisconti 1, and  Davide Catania 2,

1. 

Universitá degli Studi di Firenze, Dipartimento di Matematica e Informatica "U. Dini", Via S. Marta 3, I-50139 Firenze, Italia
2. 

SMARTest-Universitá eCampus and Universitá degli Studi di Brescia, Sezione Matematica (DICATAM), Via Valotti 9, I-25133 Brescia, Italia

Received  December 2016 Revised  January 2017 Published  May 2017

We consider the 2D simplified Bardina turbulence model, with horizontal filtering, in an unbounded strip-like domain. We prove global existence and uniqueness of weak solutions in a suitable class of anisotropic weighted Sobolev spaces.

Keywords: Simplified Bardina model, Navier-Stokes equations, turbulent flows, Large Eddy Simulation (LES), anisotropic filters, unbounded domains, global attractor.
Mathematics Subject Classification: Primary: 76D05, 35B65; Secondary: 35Q30, 76F65, 76D03.
Citation: Luca Bisconti, Davide Catania. Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1861-1881. doi: 10.3934/cpaa.2017090
References:
[1]

F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Diff. Equ., 83 (1990), 36-54.  doi: 10.1016/0022-0396(90)90070-6.

[2]

H. Ali, Approximate deconvolution model in a bounded domain with vertical regularization, J. Math. Anal. App., 408 (2013), 355-363.  doi: 10.1016/j.jmaa.2013.06.023.

[3]

C. T. Anh and P. T. Trang, On the 3D Kelvin-Voigt-Brinkman-Forchheimer equations in some unbounded domains, Nonlinear Anal., 89 (2013), 36-54.  doi: 10.1016/j.na.2013.04.014.

[4]

P. Anthony and S. Zelik, Infinite-energy solutions for the Navier-Stokes equations in a strip revisited, Commun. Pure Appl. Anal., 13 (2004), 1361-1393.  doi: 10.3934/cpaa.2014.13.1361.

[5]

A. V. Babin, The Attractor of a Navier-Stokes system in an unbounded channel-like domain, J. Dyn. Diff. Equ., 4 (1992), 555-584.  doi: 10.1007/BF01048260.

[6]

A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243.  doi: 10.1017/S0308210500031498.

[7]

J. Bardina, J. H. Ferziger and W. C. Reynolds, Improved subgrid scale models for large eddy simulation, American Institute of Aeronautics and Astronautics, 80 (1980), AIAA, 80-1357. 

[8]

L. C. Berselli, Analysis of a large eddy simulation model based on anisotropic filtering, J. Math. Anal. Appl., 386 (2012), 149-170.  doi: 10.1016/j.jmaa.2011.07.044.

[9]

L. C. Berselli and L. Bisconti, On the structural stability of the Euler-Voigt and Navier-Stokes-Voigt models, Nonlinear Anal., 75 (2012), 117-130.  doi: 10.1016/j.na.2011.08.011.

[10]

L. C. Berselli and D. Catania, On the Boussinesq equations with anisotropic filter in a vertical pipe, Dyn. Partial Differ. Equ., 12 (2015), 177-192.  doi: 10.4310/DPDE.2015.v12.n2.a5.

[11]

L. C. Berselli and D. Catania, On the well-posedness of the Boussinesq equations with anisotropic filter for turbulent flows, Z. Anal. Anwend., 34 (2015), 61-83.  doi: 10.4171/ZAA/1529.

[12]

L. Bisconti, On the convergence of an approximate deconvolution model to the 3D mean Boussinesq equations, Math. Methods Appl. Sci., 38 (2015), 1437-1450.  doi: 10.1002/mma.3160.

[13]

L. Bisconti and D. Catania, Remarks on global attractors for the 3D Navier-Stokes equations with horizontal filtering, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 59-75.  doi: 10.3934/dcdsb.2015.20.59.

[14]

L. Bisconti and P. M. Mariano, Existence results in the linear dynamics of quasicrystals with phason diffusion and nonlinear gyroscopic effects, Multiscale Model. Simul., 15 (2017), no. 2,745-767.  doi: 10.1137/15M1049580.

[15]

C. Cao, D. D. Holm and E. S. Titi, On the Clark-α model of turbulence: global regularity and long-time dynamics, J. Turbul., 6 (2005), paper 20, 11 pp.  doi: 10.1080/14685240500183756.

[16]

Y. Cao, E. M. Lunasin and E. S. Titi, Global well-posedness of three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848. 

[17]

Y. Cao and E. S. Titi, On the rate of convergence of the two-dimensional $α$-models of turbulence to the Navier-Stokes equations, Numer. Funct. Anal. Optim., 30 (2009), 1231-1271.  doi: 10.1080/01630560903439189.

[18]

D. Catania, A. Morando and P. Trebeschi, Global attractor for the Navier-Stokes equations with fractional deconvolution, Nonlinear Differ. Equ. Appl., 22 (2015), 811-848.  doi: 10.1007/s00030-014-0305-y.

[19]

A. O. Çelebi, V. K. Kalantarov and M. Polat, Global attractors for 2D Navier-Stokes-Voight equations in an unbounded domain, Appl. Anal., 88 (2009), 381-392.  doi: 10.1080/00036810902766682.

[20]

A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-α model of turbulence, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 629-649.  doi: 10.1098/rspa.2004.1373.

[21]

M. A. Efendiev and S. V. Zelik, The attractor for nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688.  doi: 10.1002/cpa.1011.

[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, J. Dynam. Differ. Equ., 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.

[23]

M. J. Garrido-Atienza and P. Mariń-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal., 64 (2006), 1100-1118.  doi: 10.1016/j.na.2005.05.057.

[24]

F. Gazzola and V. Pata, A uniform attractor for a non-autonomous generalized Navier-Stokes equation, J. Anal. Appl., 16 (1997), 435-449.  doi: 10.4171/ZAA/771.

[25]

M. Germano, Differential filters for the large eddy simulation of turbulent flows, Phys. Fluids, 29 (1986), 1755-1757.  doi: 10.1063/1.865649.

[26]

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

[27]

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

[28]

W. Layton and R. Lewandowski, On a well-posed turbulence model, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 111-128. 

[29]

W. J. Layton, C. C. Manica, M. Neda and L. G. Rebholz, Numerical analysis and computational comparisons of the NS-alpha and NS-omega regularizations, Comput. Methods Appl. Mech. Engr., 199 (2010), 916-931.  doi: 10.1016/j.cma.2009.01.011.

[30]

A. Miranville and S. Zelik, Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains, Handbook of differential equations: evolutionary equations, Vol. Ⅳ, 103-200, Handb. Differ. Equ. , Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.

[31]

M. Polat, Global attractors for a generalized 2D parabolic system in an unbounded domain, Appl. Anal., 88 (2009), 63-74.  doi: 10.1080/00036810802555508.

[32]

L. G. Rebholz, Conservation laws of turbulence models, J. Math. Anal. Appl., 326 (2007), 33-45.  doi: 10.1016/j.jmaa.2006.02.026.

[33]

J. Simon, Equations de Navier-Stokes, Cours de DEA 2002-2003, Universiteé Blaise Pascal, Clermont-Ferrand, Available from: URL http://www.lma.univ-bpclermont.fr/simon/.

[34]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[35]

S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip, Glasg. Math. J., 49 (2007), 525-588.  doi: 10.1017/S0017089507003849.

show all references

References:
[1]

F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Diff. Equ., 83 (1990), 36-54.  doi: 10.1016/0022-0396(90)90070-6.

[2]

H. Ali, Approximate deconvolution model in a bounded domain with vertical regularization, J. Math. Anal. App., 408 (2013), 355-363.  doi: 10.1016/j.jmaa.2013.06.023.

[3]

C. T. Anh and P. T. Trang, On the 3D Kelvin-Voigt-Brinkman-Forchheimer equations in some unbounded domains, Nonlinear Anal., 89 (2013), 36-54.  doi: 10.1016/j.na.2013.04.014.

[4]

P. Anthony and S. Zelik, Infinite-energy solutions for the Navier-Stokes equations in a strip revisited, Commun. Pure Appl. Anal., 13 (2004), 1361-1393.  doi: 10.3934/cpaa.2014.13.1361.

[5]

A. V. Babin, The Attractor of a Navier-Stokes system in an unbounded channel-like domain, J. Dyn. Diff. Equ., 4 (1992), 555-584.  doi: 10.1007/BF01048260.

[6]

A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243.  doi: 10.1017/S0308210500031498.

[7]

J. Bardina, J. H. Ferziger and W. C. Reynolds, Improved subgrid scale models for large eddy simulation, American Institute of Aeronautics and Astronautics, 80 (1980), AIAA, 80-1357. 

[8]

L. C. Berselli, Analysis of a large eddy simulation model based on anisotropic filtering, J. Math. Anal. Appl., 386 (2012), 149-170.  doi: 10.1016/j.jmaa.2011.07.044.

[9]

L. C. Berselli and L. Bisconti, On the structural stability of the Euler-Voigt and Navier-Stokes-Voigt models, Nonlinear Anal., 75 (2012), 117-130.  doi: 10.1016/j.na.2011.08.011.

[10]

L. C. Berselli and D. Catania, On the Boussinesq equations with anisotropic filter in a vertical pipe, Dyn. Partial Differ. Equ., 12 (2015), 177-192.  doi: 10.4310/DPDE.2015.v12.n2.a5.

[11]

L. C. Berselli and D. Catania, On the well-posedness of the Boussinesq equations with anisotropic filter for turbulent flows, Z. Anal. Anwend., 34 (2015), 61-83.  doi: 10.4171/ZAA/1529.

[12]

L. Bisconti, On the convergence of an approximate deconvolution model to the 3D mean Boussinesq equations, Math. Methods Appl. Sci., 38 (2015), 1437-1450.  doi: 10.1002/mma.3160.

[13]

L. Bisconti and D. Catania, Remarks on global attractors for the 3D Navier-Stokes equations with horizontal filtering, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 59-75.  doi: 10.3934/dcdsb.2015.20.59.

[14]

L. Bisconti and P. M. Mariano, Existence results in the linear dynamics of quasicrystals with phason diffusion and nonlinear gyroscopic effects, Multiscale Model. Simul., 15 (2017), no. 2,745-767.  doi: 10.1137/15M1049580.

[15]

C. Cao, D. D. Holm and E. S. Titi, On the Clark-α model of turbulence: global regularity and long-time dynamics, J. Turbul., 6 (2005), paper 20, 11 pp.  doi: 10.1080/14685240500183756.

[16]

Y. Cao, E. M. Lunasin and E. S. Titi, Global well-posedness of three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848. 

[17]

Y. Cao and E. S. Titi, On the rate of convergence of the two-dimensional $α$-models of turbulence to the Navier-Stokes equations, Numer. Funct. Anal. Optim., 30 (2009), 1231-1271.  doi: 10.1080/01630560903439189.

[18]

D. Catania, A. Morando and P. Trebeschi, Global attractor for the Navier-Stokes equations with fractional deconvolution, Nonlinear Differ. Equ. Appl., 22 (2015), 811-848.  doi: 10.1007/s00030-014-0305-y.

[19]

A. O. Çelebi, V. K. Kalantarov and M. Polat, Global attractors for 2D Navier-Stokes-Voight equations in an unbounded domain, Appl. Anal., 88 (2009), 381-392.  doi: 10.1080/00036810902766682.

[20]

A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-α model of turbulence, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 629-649.  doi: 10.1098/rspa.2004.1373.

[21]

M. A. Efendiev and S. V. Zelik, The attractor for nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688.  doi: 10.1002/cpa.1011.

[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, J. Dynam. Differ. Equ., 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.

[23]

M. J. Garrido-Atienza and P. Mariń-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal., 64 (2006), 1100-1118.  doi: 10.1016/j.na.2005.05.057.

[24]

F. Gazzola and V. Pata, A uniform attractor for a non-autonomous generalized Navier-Stokes equation, J. Anal. Appl., 16 (1997), 435-449.  doi: 10.4171/ZAA/771.

[25]

M. Germano, Differential filters for the large eddy simulation of turbulent flows, Phys. Fluids, 29 (1986), 1755-1757.  doi: 10.1063/1.865649.

[26]

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

[27]

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

[28]

W. Layton and R. Lewandowski, On a well-posed turbulence model, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 111-128. 

[29]

W. J. Layton, C. C. Manica, M. Neda and L. G. Rebholz, Numerical analysis and computational comparisons of the NS-alpha and NS-omega regularizations, Comput. Methods Appl. Mech. Engr., 199 (2010), 916-931.  doi: 10.1016/j.cma.2009.01.011.

[30]

A. Miranville and S. Zelik, Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains, Handbook of differential equations: evolutionary equations, Vol. Ⅳ, 103-200, Handb. Differ. Equ. , Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.

[31]

M. Polat, Global attractors for a generalized 2D parabolic system in an unbounded domain, Appl. Anal., 88 (2009), 63-74.  doi: 10.1080/00036810802555508.

[32]

L. G. Rebholz, Conservation laws of turbulence models, J. Math. Anal. Appl., 326 (2007), 33-45.  doi: 10.1016/j.jmaa.2006.02.026.

[33]

J. Simon, Equations de Navier-Stokes, Cours de DEA 2002-2003, Universiteé Blaise Pascal, Clermont-Ferrand, Available from: URL http://www.lma.univ-bpclermont.fr/simon/.

[34]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[35]

S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip, Glasg. Math. J., 49 (2007), 525-588.  doi: 10.1017/S0017089507003849.

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