Determining Modes of the 2D g-Navier-Stokes Equations

Nguyễn Đình Thi

Nội dung chính của bài viết

Tóm tắt

The “determining modes” introduced by Prodi and Foias in 1967 say that if two solutions agree asymptotically in their P projection, then they are asymptotically in their entirety (see (Foias, 1967)). We study the initial boundary value problem for 2D g-Navier-Stokes (g-NVS) equations in bounded domains with homogeneous Dirichlet boundary conditions. We find an improved upper bound on the number of deterministic modes. Moreover, we slightly improve the estimate of the number of deterministic modes and achieve the upper limit of the Grashof Gr numerical order. These estimates are consistent with heuristic estimates based on physical arguments, extends previous results by O.P. Manley and Y.M. Treve (see (Foias, 1983)). The Gronwall lemma and Poincaré type inequality will play a central role in our computational technique as well as  of the paper. Studying the properties of solutions is important to determine the behavior of solutions over a long period of time. The obtained result particularly extends previous results for 2D NVS equations.

Chi tiết bài viết

Tài liệu tham khảo

Catania, D. (2012). Finite dimensional global attractor for 3D MHD-α models: a comparison. Journal of Mathematical Fluid Mechanics, 14, 95-115.
Foias, C., Manley, O. P., Temam, R., & Treve, Y. M. (1983). Asymptotic analysis of the Navier-Stokes equations. Physica D: Nonlinear Phenomena, 9(1-2), 157-188.
Foias, C., Manley, O., & Temam, R. (1987). Attractors for the Bénard problem: existence and physical bounds on their fractal dimension. Nonlinear Analysis: Theory, Methods & Applications, 11(8), 939-967.
Foias, C., & Prodi, G. (1967). Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2. Rendiconti del Seminario matematico della Universita di Padova, 39, 1-34.
Foias, C., & Temam, R. (1984). Determination of the solutions of the Navier-Stokes equations by a set of nodal values. Mathematics of Computation, 43(167), 117-133.
Jones, D. A., & Titi, E. S. (1993). Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations. Indiana University Mathematics Journal, 875-887.
Olson, E., & Titi, E. S. (2003). Determining modes for continuous data assimilation in 2D turbulence. Journal of statistical physics, 113, 799-840.
Olson, E., & Titi, E. S. (2008). Determining modes and Grashof number in 2D turbulence: a numerical case study. Theoretical and Computational Fluid Dynamics, 22, 327-339.
Özlük, M., & Meryem, K. A. Y. A. (2016). On the weak solutions and determining modes of the g-Bénard problem. Hacettepe Journal of Mathematics and Statistics, 47(6), 1453-1466.