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Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc
Remarks on global attractors for the 3D Navier--Stokes equations with horizontal filtering
1. | Università degli Studi di Firenze, Dipartimento di Matematica e Informatica "U. Dini", Via S. Marta 3, I-50139 Firenze, Italy |
2. | Università degli Studi di Brescia, Sezione Matematica (DICATAM), Via Valotti 9, I-25133 Brescia, Italy |
  It is known that there exists a unique regular weak solution to this model that depends weakly continuously on the initial datum. We show the existence of the global attractor for the semiflow given by the time-shift in the space of paths. We prove the continuity of the horizontal components of the flow under periodicity in all directions and discuss the possibility to introduce a solution semiflow.
References:
[1] |
N. A. Adams and S. Stolz, Deconvolution methods for subgrid-scale approximation in large eddy simulation, in Modern Simulation Strategies for Turbulent Flow (ed. B. J. Geurts), R. T. Edwards, 2001, 21-44. |
[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] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. |
[4] |
L. C. Berselli, T. Iliescu and W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Springer, 2006. |
[5] |
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. |
[6] |
L. C. Berselli and D. Catania, On the well-posedness of the Boussinesq equations with horizontal filter for turbulent flows, to appear in Z. Anal. Anwend. |
[7] |
L. C. Berselli and D. Catania, On the equations with anisotropic filter in a vertical pipe, to appear in Dyn. Partial Differ. Equ. |
[8] |
L. C. Berselli, D. Catania and R. Lewandowski, Convergence of approximate deconvolution models to the mean magnetohydrodynamics equations: Analysis of two models, J. Math. Anal. Appl., 401 (2013), 864-880.
doi: 10.1016/j.jmaa.2012.12.051. |
[9] |
H. Bessaih and F. Flandoli, Weak attractor for a dissipative Euler equation, J. Dynamics Diff. Equations, 12 (2000), 713-732.
doi: 10.1023/A:1009042520953. |
[10] |
L. Bisconti, On the convergence of an approximate deconvolution model to the 3D mean Boussinesq equations, to appear in Mathematical Methods in the Applied Sciences, published online, (2014).
doi: 10.1002/mma.3160. |
[11] |
D. Catania and P. Secchi, Global existence and finite dimensional global attractor for a 3D double viscous MHD-$\alpha$ model, Commun. Math. Sci., 8 (2010), 1021-1040.
doi: 10.4310/CMS.2010.v8.n4.a12. |
[12] |
J. W. Deardorff, A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers, J. Fluid Mech., 41 (1970), 453-480.
doi: 10.1017/S0022112070000691. |
[13] |
C. Foias, 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. Differential Equations, 14 (2002), 1-35.
doi: 10.1023/A:1012984210582. |
[14] |
M. Germano, Differential filters for the large eddy simulation of turbulent flows, Phys. Fluids, 29 (1986), 1755-1757.
doi: 10.1063/1.865649. |
[15] |
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. |
[16] |
J. L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications, Dunod, Paris, 1968. |
[17] |
G. R. Sell, Global attractor for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33.
doi: 10.1007/BF02218613. |
[18] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[19] |
A. Scotti, C. Meneveau and D. K. Lilly, Generalized Smagorinsky model for anisotropic grids, Phys. Fluids, 5 (1993), 2306.
doi: 10.1063/1.858537. |
[20] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer, 1988.
doi: 10.1007/978-1-4684-0313-8. |
show all references
References:
[1] |
N. A. Adams and S. Stolz, Deconvolution methods for subgrid-scale approximation in large eddy simulation, in Modern Simulation Strategies for Turbulent Flow (ed. B. J. Geurts), R. T. Edwards, 2001, 21-44. |
[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] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. |
[4] |
L. C. Berselli, T. Iliescu and W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Springer, 2006. |
[5] |
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. |
[6] |
L. C. Berselli and D. Catania, On the well-posedness of the Boussinesq equations with horizontal filter for turbulent flows, to appear in Z. Anal. Anwend. |
[7] |
L. C. Berselli and D. Catania, On the equations with anisotropic filter in a vertical pipe, to appear in Dyn. Partial Differ. Equ. |
[8] |
L. C. Berselli, D. Catania and R. Lewandowski, Convergence of approximate deconvolution models to the mean magnetohydrodynamics equations: Analysis of two models, J. Math. Anal. Appl., 401 (2013), 864-880.
doi: 10.1016/j.jmaa.2012.12.051. |
[9] |
H. Bessaih and F. Flandoli, Weak attractor for a dissipative Euler equation, J. Dynamics Diff. Equations, 12 (2000), 713-732.
doi: 10.1023/A:1009042520953. |
[10] |
L. Bisconti, On the convergence of an approximate deconvolution model to the 3D mean Boussinesq equations, to appear in Mathematical Methods in the Applied Sciences, published online, (2014).
doi: 10.1002/mma.3160. |
[11] |
D. Catania and P. Secchi, Global existence and finite dimensional global attractor for a 3D double viscous MHD-$\alpha$ model, Commun. Math. Sci., 8 (2010), 1021-1040.
doi: 10.4310/CMS.2010.v8.n4.a12. |
[12] |
J. W. Deardorff, A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers, J. Fluid Mech., 41 (1970), 453-480.
doi: 10.1017/S0022112070000691. |
[13] |
C. Foias, 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. Differential Equations, 14 (2002), 1-35.
doi: 10.1023/A:1012984210582. |
[14] |
M. Germano, Differential filters for the large eddy simulation of turbulent flows, Phys. Fluids, 29 (1986), 1755-1757.
doi: 10.1063/1.865649. |
[15] |
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. |
[16] |
J. L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications, Dunod, Paris, 1968. |
[17] |
G. R. Sell, Global attractor for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33.
doi: 10.1007/BF02218613. |
[18] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[19] |
A. Scotti, C. Meneveau and D. K. Lilly, Generalized Smagorinsky model for anisotropic grids, Phys. Fluids, 5 (1993), 2306.
doi: 10.1063/1.858537. |
[20] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer, 1988.
doi: 10.1007/978-1-4684-0313-8. |
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