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

Abstract

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.