Estimates for the Dimension of Attractors of a Regularized Euler System with Dissipation on the Sphere

S. V. Zelik ; A. A. Ilyin ; A. G. Kostyanko

Matematičeskie zametki, Tome 111 (2022) no. 1, pp. 54-66.

Résumé

We prove the existence of a global attractor of a regularized Euler–Bardina system with dissipation on the two-dimensional sphere and in arbitrary domains on the sphere. Explicit estimates for the fractal dimension of the attractor in terms of its physical parameters are obtained.

Détail

Keywords: Euler's system, Bardina's model, attractors, spectral inequalities on the sphere.
Mots-clés : fractal dimension

Bibliographie

[1] J. Bardina, J. Ferziger, W. Reynolds, “Improved subgrid scale models for large eddy simulation”, Proceedings of the 13th AIAA Conference on Fluid and Plasma Dynamics, 1980 | DOI

[2] Y. Cao, E. M. Lunasin, E. S. Titi, “Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models”, Commun. Math. Sci., 4:4 (2006), 823–848 | DOI | Zbl

[3] V. K. Kalantarov, E. S. Titi, “Global attractors and determining modes for the 3D Navier–Stokes–Voight equations”, Chin. Ann. Math., 30B:6 (2009), 697–714 | DOI | Zbl

[4] A. A. Ilyin, S. V. Zelik, “Sharp dimension estimates of the attractor of the damped 2D Euler–Bardina equations”, Partial Differential Equations, Spectral Theory, and Mathematical Physics, EMS Series of Congress Reports, 18, EMS Press, Berlin, 2021, 209–229

[5] A. A. Ilyin, A. G. Kostianko, S. V. Zelik, Sharp Upper and Lower Bounds of the Attractor Dimension for 3D Damped Euler–Bardina Equations, 2021, arXiv: 2106.09077

[6] S. V. Zelik, A. A. Ilin, A. G. Kostyanko, “Tochnye otsenki razmernosti attraktorov trekhmernoi regulyarizirovannoi sistemy Eilera s dissipatsiei”, Dokl. AN, 499:1 (2021), 13–16 | DOI | Zbl

[7] A. A. Ilin, “Uravneniya Nave–Stoksa i Eilera na dvumernykh zamknutykh mnogoobraziyakh”, Matem. sb., 181:4 (1990), 521–539 | MR | Zbl

[8] E. H. Lieb, “An $L^p$ bound for the Riesz and Bessel potentials of orthonormal functions”, J. Func. Anal., 51 (1983), 159–165 | DOI | Zbl

[9] B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, Sovremennaya geometriya. Metody i prilozheniya, Nauka, M., 1979 | MR | Zbl

[10] A. V. Babin, M. I. Vishik, Attraktory evolyutsionnykh uravnenii, Nauka, M., 1989 | MR | Zbl

[11] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997 | MR | Zbl

[12] A. A. Ilyin, A. Miranville, 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 | DOI | Zbl

[13] V. V. Chepyzhov, A. A. Ilyin, “A note on the fractal dimension of attractors of dissipative dynamical systems”, Nonlinear Anal., 44 (2001), 811–819 | DOI | Zbl

[14] V. V. Chepyzhov, A. A. Ilyin, “On the fractal dimension of invariant sets: applications to Navier–Stokes equations”, Discrete Contin. Dyn. Syst., 10:1-2 (2004), 117–135 | MR | Zbl

[15] E. Lieb, W. Thirring, “Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities”, Studies in Mathematical Physics. Essays in honor of Valentine Bargmann, Princeton Univ. Press, Princeton, NJ, 1976, 269–303

[16] H. Araki, “On an inequality of Lieb and Thirring”, Lett. Math. Phys., 19 (1990), 167–170 | DOI | Zbl

[17] B. Simon, Trace Ideals and Their Applications, Amer. Math. Soc., Providence RI, 2005 | MR | Zbl

[18] S. V. Zelik, A. A. Ilin, “Asimptotika funktsii Grina i tochnye interpolyatsionnye neravenstva”, UMN, 69:2 (416) (2014), 23–76 | DOI | MR | Zbl

[19] A. A. Ilyin, “Best constants in Sobolev inequalities on the sphere and in Euclidean space”, J. London Math. Soc. (2), 59 (1999), 263–286 | DOI | Zbl

[20] A. A. Ilyin, “Lieb–Thirring inequalities on some manifolds”, J. Spectr. Theory, 2 (2012), 57–78 | DOI | Zbl