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.

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We consider the 2D g-Bénard problem in domains satisfying the Poincaré inequality with homogeneous Dirichlet boundary conditions. We prove the existence and uniqueness of global weak solutions. The obtained results particularly extend previous results for 2D g-Navier-Stokes equations and 2D Bénard problem.