Sharp dimension estimates of the attractor of the damped 2D Euler-Bardina equations. (English) Zbl 1479.35101

Exner, Pavel (ed.) et al., Partial differential equations, spectral theory, and mathematical physics. The Ari Laptev anniversary volume. Berlin: European Mathematical Society (EMS). EMS Ser. Congr. Rep., 209-229 (2021).
Summary: We prove existence of the global attractor of the damped and driven 2D Euler-Bardina equations on the torus and give an explicit two-sided estimate of its dimension that is sharp as \(\alpha\to 0^+\).
For the entire collection see [Zbl 1465.35005].


35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
35B45 A priori estimates in context of PDEs
35Q31 Euler equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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[1] H. Araki, On an inequality of Lieb and Thirring. Lett. Math. Phys. 19 (1990), 167-170 · Zbl 0705.47020
[2] A. V. Babin and M. I. Vishik, Attractors of evolution equations. Stud. Math. Appl.25, North-Holland, Amsterdam, 1992 · Zbl 0778.58002
[3] J. Bardina, J. Ferziger and W. Reynolds, Improved subgrid scale models for large eddy simulation. AIAA paper 80-1357, 1980
[4] Y. Cao, E. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models. Commun. Math. Sci. 4 (2006), 823-848 · Zbl 1127.35034
[5] V. V. Chepyzhov and A. A. Ilyin, A note on the fractal dimension of attractors of dissipative dynamical systems. Nonlinear Anal. 44 (2001), 811-819 · Zbl 1153.37438
[6] V. V. Chepyzhov and A. A. Ilyin, On the fractal dimension of invariant sets; applications to Navier-Stokes equations. Discrete Contin. Dyn. Syst. 10 (2004), 117-135 · Zbl 1049.37047
[7] V. V. Chepyzhov, A. A. Ilyin and S. Zelik, Vanishing viscosity limit for global attractors for the damped Navier-Stokes system with stress free boundary conditions. Phys. D 376/377 (2018), 31-38 · Zbl 1398.35145
[8] V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics. Amer. Math. Soc. Colloq. Publ. 49, American Mathematical Society, Providence, RI, 2002 · Zbl 0986.35001
[9] V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for dissipative 2D Euler and Navier-Stokes equations. Russ. J. Math. Phys. 15 (2008), 156-170 · Zbl 1180.35420
[10] V. V. Chepyzhov, M. I. Vishik and S. V. Zelik, Strong trajectory attractors for dissipative Euler equations. J. Math. Pures Appl. (9) 96 (2011), 395-407 · Zbl 1230.35092
[11] P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations. Comm. Pure Appl. Math. 38 (1985), 1-27 · Zbl 0582.35092
[12] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes equations and turbulence. Encyclopedia Math. Appl. 83, Cambridge University Press, Cambridge, 2001 · Zbl 0994.35002
[13] A. Haraux, Two remarks on hyperbolic dissipative problems. In Nonlinear partial dif-ferential equations and their applications. Collège de France seminar, Vol. VII (Paris, 1983-1984), pp. 6, 161-179, Res. Notes in Math. 122, Pitman, Boston, 1985 · Zbl 0579.35057
[14] A. A. Ilyin, Euler equations with dissipation. Mat. Sb. 182 (1991), 1729-1739 (in Russian); · Zbl 0766.35038
[15] A. A. Ilyin, A. G. Kostianko and S. V. Zelik, Finite-dimensional attractors for damped Euler-Bardina model in three dimensions. In preparation
[16] A. A. Ilyin, A. Miranville and E. S. Titi, Small viscosity sharp estimates for the global attractor of the 2-D damped-driven Navier-Stokes equations. Commun. Math. Sci. 2 (2004), 403-426 · Zbl 1084.35058
[17] A. A. Ilyin and A. A. Laptev, Lieb-Thirring inequalities on the torus. Mat. Sb. 207 (2016), 56-79 (in Russian); · Zbl 1365.35096
[18] English transl. Sb. Math. 207 (2016), 1410-1434
[19] A. A. Ilyin, A. A. Laptev and S. Zelik, Lieb-Thirring constant on the sphere and on the torus. J. Funct. Anal. 279 (2020), Article ID 108784 · Zbl 1450.35196
[20] A. A. Ilyin and E. S. Titi, Attractors for the two-dimensional Navier-Stokes-˛model: An -dependence study. J. Dynam. Differential Equations 15 (2003), 751-778 · Zbl 1039.35078
[21] W. B. Jones and W. J. Thron, Continued fractions. Analytic theory and applications. Encyclopedia Math. Appl. 11, Addison-Wesley Publishing, Reading, 1980 · Zbl 0445.30003
[22] V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations. Chin. Ann. Math. Ser. B 30 (2009), 697-714 · Zbl 1178.37112
[23] O. Ladyzhenskaya, Attractors for semigroups and evolution equations. Lezioni Lincee, Cambridge University Press, Cambridge, 1991
[24] E. Lieb, An L p bound for the Riesz and Bessel potentials of orthonormal functions. J. Funct. Anal. 51 (1983), 159-165 · Zbl 0517.46025
[25] E. Lieb, On characteristic exponents in turbulence. Comm. Math. Phys. 92 (1984), 473-480 · Zbl 0598.76054
[26] E. Lieb and W. Thirring, Inequalities for the moments of the eigenvalues of the Schrö-dinger Hamiltonian and their relation to Sobolev inequalities. Studies in Mathematical Physics. Essays in honor of Valentine Bargmann, pp. 269-303, Princeton University Press, Princeton, 1976
[27] V. X. Liu, A sharp lower bound for the Hausdorff dimension of the global attractors of the 2D Navier-Stokes equations. Comm. Math. Phys. 158 (1993), 327-339 · Zbl 0790.35085
[28] L. D. Meshalkin and Ya. G. Sinai, Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid. Prikl. Mat. Mekh. 25 (1961), 1140-1143 (in Russian); · Zbl 0108.39501
[29] English transl. J. Appl. Math. Mech. 25 (1961), 1700-1705
[30] J. Pedlosky, Geophysical fluid dynamics. Springer, New York, 1979 · Zbl 0429.76001
[31] G. R. Sell and Y. You, Dynamics of evolutionary equations. Appl. Math. Sci. 143, Springer, New York, 2002 · Zbl 1254.37002
[32] B. Simon, Trace ideals and their applications. 2nd edn., Math. Surveys Monogr. 120, American Mathematical Society, Providence, 2005 · Zbl 1074.47001
[33] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, 1971 · Zbl 0232.42007
[34] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics. 2nd edn., Appl. Math. Sci. 68, Springer, New York, 1997 · Zbl 0871.35001
[35] G. N. Watson, A treatise on the theory of Bessel functions. Cambridge Math. Libr., Cam-bridge University Press, Cambridge, 1995 · Zbl 0849.33001
[36] Z.-H. Yang and Y.-M. Chu, On approximating the modified Bessel function of the second kind. J. Inequal. Appl. 41 (2017), Paper No. 41
[37] V. I. Yudovich, Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid. Prikl. Mat. Mekh. 29 (1965), 453-467 (in Russian); · Zbl 0148.22307
[38] English transl. J. Appl. Math. Mech. 29 (1965
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