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

Article Contents

2017, Volume 16, Issue 5: 1861-1881. Doi: 10.3934/cpaa.2017090

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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: October 2017

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Abstract

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:

Mathematics Subject Classification: Primary: 76D05, 35B65; Secondary: 35Q30, 76F65, 76D03.

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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.