Upper bound on the number of determining modes for the 2D g-bénard problem
Email:
ndthi.spktnd@moet.edu.vn
Từ khóa:
g-Bénard problem, determining modes, Grashof number
Tóm tắt
The "determining modes" concept introduced by Foias and Prodi in 1967 say that if two solutions agree asymptotically in their P projection, then they are asymptotical in their entirety. In this paper, we consider the 2D g-Bénard problem in domains satisfying the Poincaré inequality with homogeneous Dirichlet boundary conditions. We present an improved upper bound on the number of determining modes. Moreover, we slightly improve the estimate on the number of determining modes and obtain an upper bound of the order G. These estimates are in agreement with the heuristic estimates based on physical arguments, that have been conjectured by O.P. Manley and Y.M. Treve. The Gronwall lemma and Poincaré type inequality will play a central role in our computational technique as well as the proof of the main result of the paper. Studying the properties of solutions is important to determine the behavior of solutions over a long period of time. The obtained results particularly extend previous results for 2D g-Navier-Stokes equations and 2D Bénard problem.Tài liệu tham khảo
[1]. Davide Catania, Finite Dimensional Global Attractor for 3D MHD-α Models: A Comparison, J. Math. Fluid Mech, 14 (2012) 95-115. https://doi.org/10.1007/s00021-010-0041-y
[2]. C. Foias, O. Manley, R. Temam, Y. Treve, Asymptotic analysis of the Navier-Stokes equations, Phys. D, 9 (1983) 157-188. https://doi.org/10.1016/0167-2789(83)90297-X
[3]. C. Foias, O. Manley, R. Temam, Attractors for the B´enard problem: Existence and physical bounds on their fractal dimension, Nonlinear Analysis: Theory, Methods & Applications, 11 (1987) 939-967. https://doi.org/10.1016/0362-546X(87)90061-7
[4]. C. Foias, G. Prodi, Sur le comportement global des solutions non stationnaires des équations de Navier-Stokes en dimension two, Rend. Sem. Mat. Univ., Padova, 39 (1967) 1-34.
[5]. C. Foias, R. Temam, Determination of the solutions of the Navier-Stokes equations by
a set of nodal values, Math. Comput., 43 (1984) 117-133. https://doi.org/10.2307/2007402
[6]. D. Jones, E.S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier - Stokes equations, Indiana Univ. Math. J., 42 (1993) 875-887.
[7]. E. Olson, E.S. Titi, Determining modes for continuous data assimilation in 2D turbulence, Journal of Statistical Physics, 113 (2003) 799-840. https://doi.org/10.1023/A:1027312703252
[8]. E. Olson, E.S. Titi, Determining modes and Grashof number in 2D turbulence: a numerical case study, Theoretical and Computational Fluid Dynamics, 22 (2008) 327-339. https://doi.org/10.1007/s00162-008-0086-1
[9]. M. Ozluk, M. Kaya, On the weak solutions and determining modes of the g-Bénard problem, Hacet. J. Math. Stat., 47 (2018) 1453-1466. https://doi.org/10.15672/HJMS.20174622762
[10]. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS Regional Conference Series, No. 41, SIAM, Philadelphia, 1983.
[11]. R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, vol. 2 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 3rd edition, 1984.
[12]. J. Roh, Dynamics of the g-Navier-Stokes equations, Journal of Differential Equations, 211 (2005) 452-484. https://doi.org/10.1016/j.jde.2004.08.016
[2]. C. Foias, O. Manley, R. Temam, Y. Treve, Asymptotic analysis of the Navier-Stokes equations, Phys. D, 9 (1983) 157-188. https://doi.org/10.1016/0167-2789(83)90297-X
[3]. C. Foias, O. Manley, R. Temam, Attractors for the B´enard problem: Existence and physical bounds on their fractal dimension, Nonlinear Analysis: Theory, Methods & Applications, 11 (1987) 939-967. https://doi.org/10.1016/0362-546X(87)90061-7
[4]. C. Foias, G. Prodi, Sur le comportement global des solutions non stationnaires des équations de Navier-Stokes en dimension two, Rend. Sem. Mat. Univ., Padova, 39 (1967) 1-34.
[5]. C. Foias, R. Temam, Determination of the solutions of the Navier-Stokes equations by
a set of nodal values, Math. Comput., 43 (1984) 117-133. https://doi.org/10.2307/2007402
[6]. D. Jones, E.S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier - Stokes equations, Indiana Univ. Math. J., 42 (1993) 875-887.
[7]. E. Olson, E.S. Titi, Determining modes for continuous data assimilation in 2D turbulence, Journal of Statistical Physics, 113 (2003) 799-840. https://doi.org/10.1023/A:1027312703252
[8]. E. Olson, E.S. Titi, Determining modes and Grashof number in 2D turbulence: a numerical case study, Theoretical and Computational Fluid Dynamics, 22 (2008) 327-339. https://doi.org/10.1007/s00162-008-0086-1
[9]. M. Ozluk, M. Kaya, On the weak solutions and determining modes of the g-Bénard problem, Hacet. J. Math. Stat., 47 (2018) 1453-1466. https://doi.org/10.15672/HJMS.20174622762
[10]. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS Regional Conference Series, No. 41, SIAM, Philadelphia, 1983.
[11]. R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, vol. 2 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 3rd edition, 1984.
[12]. J. Roh, Dynamics of the g-Navier-Stokes equations, Journal of Differential Equations, 211 (2005) 452-484. https://doi.org/10.1016/j.jde.2004.08.016
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Nhận bài
06/04/2022
Nhận bài sửa
30/05/2022
Chấp nhận đăng
10/06/2022
Xuất bản
15/09/2022
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Kiểu trích dẫn
Nguyen Dinh, T., Tran Quang, T., & Trinh The, A. (1663174800). Upper bound on the number of determining modes for the 2D g-bénard problem. Tạp Chí Khoa Học Giao Thông Vận Tải, 73(7), 674-687. https://doi.org/10.47869/tcsj.73.7.2
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