-
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
Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region
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 |
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.
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. |
[1] |
Daniel Coutand, Steve Shkoller. Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 1-23. doi: 10.3934/cpaa.2004.3.1 |
[2] |
Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234 |
[3] |
Reinhard Farwig, Yasushi Taniuchi. Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1215-1224. doi: 10.3934/dcdss.2013.6.1215 |
[4] |
Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209 |
[5] |
Ciprian Foias, Ricardo Rosa, Roger Temam. Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1611-1631. doi: 10.3934/dcds.2010.27.1611 |
[6] |
Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101 |
[7] |
Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure and Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35 |
[8] |
Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613 |
[9] |
Qi S. Zhang. An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5521-5523. doi: 10.3934/dcds.2013.33.5521 |
[10] |
Siegfried Maier, Jürgen Saal. Stokes and Navier-Stokes equations with perfect slip on wedge type domains. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 1045-1063. doi: 10.3934/dcdss.2014.7.1045 |
[11] |
Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299 |
[12] |
Alain Miranville, Xiaoming Wang. Upper bound on the dimension of the attractor for nonhomogeneous Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 1996, 2 (1) : 95-110. doi: 10.3934/dcds.1996.2.95 |
[13] |
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 |
[14] |
Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355 |
[15] |
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 |
[16] |
Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783 |
[17] |
Reinhard Farwig, Paul Felix Riechwald. Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 157-172. doi: 10.3934/dcdss.2016.9.157 |
[18] |
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 |
[19] |
Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045 |
[20] |
Shuguang Shao, Shu Wang, Wen-Qing Xu. Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation. Kinetic and Related Models, 2018, 11 (1) : 179-190. doi: 10.3934/krm.2018009 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]