Trajectory attractors for 3D damped Euler equations and their approximation. (English) Zbl 1500.35051
Summary: We study the global attractors for the damped 3D Euler-Bardina equations with the regularization parameter \(\alpha>0\) and Ekman damping coefficient \(\gamma>0\) endowed with periodic boundary conditions as well as their damped Euler limit \(\alpha\to 0\). We prove that despite the possible non-uniqueness of solutions of the limit Euler system and even the non-existence of such solutions in the distributional sense, the limit dynamics of the corresponding dissipative solutions introduced by P. Lions can be described in terms of attractors of the properly constructed trajectory dynamical system. Moreover, the convergence of the attractors \(\mathcal{A}(\alpha)\) of the regularized system to the limit trajectory attractor \(\mathcal{A}(0)\) as \(\alpha\to 0\) is also established in terms of the upper semicontinuity in the properly defined functional space.
References:
| [1] | A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, vol 25. North-Holland Publishing Co., Amsterdam, 1992. · Zbl 0778.58002 |
| [2] | J. Bardina, J. Ferziger and W. Reynolds, Improved Subgrid Scale Models for Large Eddy Simulation, in Proceedings of the 13th AIAA Conference on Fluid and Plasma Dynamics, 1980. |
| [3] | K. Bardos; E. Titi, Euler equations for incompressible ideal fluids, Uspekhi Mat. Nauk, 62, 5-46 (2007) · Zbl 1139.76010 · doi:10.1070/RM2007v062n03ABEH004410 |
| [4] | Y. Cao; E. Lunasin; E. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4, 823-848 (2006) · Zbl 1127.35034 · doi:10.4310/CMS.2006.v4.n4.a8 |
| [5] | V. Chepyzhov; A. Ilyin; S. Zelik, Vanishing viscosity limit for global attractors for the damped Navier-Stokes system with stress free boundary conditions, Physica D, 376-377, 31-38 (2018) · Zbl 1398.35145 · doi:10.1016/j.physd.2017.08.005 |
| [6] | V. Chepyzhov; A. Ilyin; S. Zelik, Strong trajectory and global \(W^{1, p}\)-attractors for the damped-driven Euler system in \({\mathbb R}^2\), Discrete Contin. Dyn. Syst. Ser. B, 22, 1835-1855 (2017) · Zbl 1359.35013 · doi:10.3934/dcdsb.2017109 |
| [7] | V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., 49, Providence, RI: Amer. Math. Soc., 2002. · Zbl 0986.35001 |
| [8] | V. Chepyzhov; M. Vishik; S. Zelik, Strong trajectory attractors for the dissipative Euler equations, J. Math. Pures Appl., 96, 395-407 (2011) · Zbl 1230.35092 · doi:10.1016/j.matpur.2011.04.007 |
| [9] | V. Chepyzhov; S. Zelik, Infinite energy solutions for dissipative Euler equations in \({\mathbb R}^2\), J. Math. Fluid Mech., 17, 513-532 (2015) · Zbl 1325.35137 · doi:10.1007/s00021-015-0213-x |
| [10] | R. DiPerna; A. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108, 667-689 (1987) · Zbl 0626.35059 · doi:10.1007/BF01214424 |
| [11] | C. Fefferman, Existence and smoothness of the Navier-Stokes equation, In: Millennium Prize Problems, Clay Math. Inst., Cambridge, MA, (2006), 57-67. · Zbl 1194.35002 |
| [12] | C. Foias; D. Holm; E. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differential Equations, 14, 1-35 (2002) · Zbl 0995.35051 · doi:10.1023/A:1012984210582 |
| [13] | M. Holst; E. Lunasin; G. Tsogtgerel, Analysis of a general family of regularized Navier-Stokes and MHD models, J. Nonlinear Sci., 20, 523-567 (2010) · Zbl 1207.35241 · doi:10.1007/s00332-010-9066-x |
| [14] | A. Il’in, The Euler equations with dissipation, Mat. Sb., 74, 475-485 (1993) · Zbl 0774.35057 · doi:10.1070/SM1993V074N02ABEH003357 |
| [15] | A. Ilyin, A. Kostianko and S. Zelik, Sharp upper and lower bounds of the attractor dimension for 3D damped Euler-Bardina equations, Physica D, 432 (2022), Paper No. 133156. · Zbl 1490.35273 |
| [16] | A. Ilyin; A. Miranville; E. Titi, Small viscosity sharp estimates for the global attractor of the 2D damped-driven Navier-Stokes equations, Commun. Math. Sci., 2, 403-426 (2004) · Zbl 1084.35058 · doi:10.4310/CMS.2004.v2.n3.a4 |
| [17] | A. Ilyin; E. Titi, Attractors to the two-dimensional Navier-Stokes-\( \alpha\) models: An \(\alpha \)-dependence study, J. Dynam. Differential Equations, 15, 751-778 (2003) · Zbl 1039.35078 · doi:10.1023/B:JODY.0000010064.06851.ff |
| [18] | A. Ilyin and S. Zelik, Sharp dimension estimates of the attractor of the damped 2D Euler-Bardina equations, In book: Partial Differential Equations, Spectral Theory, and Mathematical Physics, EMS Series of Congress Reports, EMS Press, Berlin, (2021), 209-229. · Zbl 1479.35101 |
| [19] | V. Kalantarov; E. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. Ser. B, 30, 697-714 (2009) · Zbl 1178.37112 · doi:10.1007/s11401-009-0205-3 |
| [20] | W. Layton; R. Lewandowski, On a well-posed turbulence model, Discrete Continuous Dyn. Sys. B, 6, 111-128 (2006) · Zbl 1089.76028 · doi:10.3934/dcdsb.2006.6.111 |
| [21] | J. Leray, Sur le mouvement d’un fluide visqueux emplissant l’espace, Acta Math., 63, 193-248 (1934) · JFM 60.0726.05 · doi:10.1007/BF02547354 |
| [22] | J.-L. Lions, Quelques Méthodes des Problèmes aux Limites non Linéaires, Doud, Paris, 1969. · Zbl 0189.40603 |
| [23] | P.-L. Lions, Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models, Oxford Lecture Series in Mathematics and Its Applications, 1996. · Zbl 0866.76002 |
| [24] | M. Lopes Filho; H. Nussenzveig Lopes; E. Titi; A. Zang, Convergence of the 2D Euler-\( \alpha\) to Euler equations in the Dirichlet case: Indifference to boundary layers, Phys. D, 292-293, 51-61 (2015) · Zbl 1364.35277 · doi:10.1016/j.physd.2014.11.001 |
| [25] | A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, In: Handbook of Differential Equations: Evolutionary Equations. Vol. IV, 103-200, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. · Zbl 1221.37158 |
| [26] | E. Olson; E. Titi, Viscosity versus vorticity stretching: Global well-posedness for a family of Navier-Stokes-\( \alpha \)-like models, Nonlinear Anal., 66, 2427-2458 (2007) · Zbl 1110.76011 · doi:10.1016/j.na.2006.03.030 |
| [27] | A. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Nauchn. Sem. LOMI, 38, 98-136 (1973) |
| [28] | J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1987. · Zbl 0713.76005 |
| [29] | T. Tao, Finite time blowup for an averaged three-dimensional Navier-Stokes equation, J. Amer. Math. Soc., 29, 601-674 (2016) · Zbl 1342.35227 · doi:10.1090/jams/838 |
| [30] | R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Springer-Verlag, New York 1997. · Zbl 0871.35001 |
| [31] | R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, vol. 66, Siam, 1995. · Zbl 0833.35110 |
| [32] | E. Wiedemann, Existence of weak solutions for the incompressible Euler equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28, 727-730 (2011) · Zbl 1228.35172 · doi:10.1016/j.anihpc.2011.05.002 |
| [33] | V. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. Res. Lett., 2, 27-38 (1995) · Zbl 0841.35092 · doi:10.4310/MRL.1995.v2.n1.a4 |
| [34] | S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, Disc. Cont. Dyn. Sys., 11, 351-392 (2004) · Zbl 1059.35018 · doi:10.3934/dcds.2004.11.351 |
| [35] | S. Zelik; A. Ilyin; A. Kostianko, Sharp dimension estimates for the attractors of the regularized damped Euler system, Doklady Mathematics, 104, 169-172 (2021) · Zbl 1477.35149 · doi:10.1134/S1064562421040165 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
