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

Luca Bisconti; Davide Catania

doi:10.3934/cpaa.2017090

Previous Article
Global dynamics of a microorganism flocculation model with time delay
CPAA Home
This Issue
Next Article
Optimality conditions of the first eigenvalue of a fourth order Steklov problem

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 & 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	M. Germano, Differential filters for the large eddy simulation of turbulent flows, Phys. Fluids, 29 (1986), 1755-1757. doi: 10.1063/1.865649. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	W. Layton and R. Lewandowski, On a well-posed turbulence model, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 111-128. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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/. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	M. Germano, Differential filters for the large eddy simulation of turbulent flows, Phys. Fluids, 29 (1986), 1755-1757. doi: 10.1063/1.865649. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	W. Layton and R. Lewandowski, On a well-posed turbulence model, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 111-128. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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/. Google Scholar
[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. Google Scholar
[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. Google Scholar

[1]	Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[2]	Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020348
[3]	Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242
[4]	Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020345
[5]	Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268
[6]	Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267
[7]	Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082
[8]	Andreu Ferré Moragues. Properties of multicorrelation sequences and large returns under some ergodicity assumptions. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020386
[9]	Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122
[10]	Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
[11]	Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240
[12]	Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020460
[13]	Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119
[14]	Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[15]	Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020457
[16]	Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020316
[17]	Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020317
[18]	Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219
[19]	Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020344
[20]	Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020349

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors