We consider the 2D g-B´enard problem in domains satisfying the Poincar´e 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 B´enard problem. | HNUE JOURNAL OF SCIENCE DOI Natural Science 2020 Volume 65 Issue 6 pp. 23-31 This paper is available online at http ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS TO 2D G-BENARD PROBLEM IN UNBOUNDED DOMAINS Tran Quang Thinh1 and Le Thi Thuy2 1 Faculty of Basic Sciences Nam Dinh University of Technology Education 2 Faculty of Mathematics Electric Power University Abstract. We consider the 2D g-B enard problem in domains satisfying the Poincar e 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 B enard problem. 1. Introduction Let Ω be a not necessarily bounded domain in R2 with boundary Γ. We consider the following two-dimensional 2D g-B enard problem u u u ν u p ξθ f1 x Ω t gt 0 t gu 0 x Ω t gt 0 θ 2κ κ g u θ κ θ g θ θ f2 x Ω t gt 0 t g g u 0 x Γ t gt 0 θ 0 x Γ t gt 0 u x 0 u0 x x Ω θ x 0 θ0 x x Ω where u u x t u1 u2 is the unknown velocity vector θ θ x t is the temperature p p x t is the unknown pressure f1 is the external force function f2 is the heat source function ν gt 0 is the kinematic viscosity coefficient ξ is a constant vector κ gt 0 is thermal diffusivity u0 is the initial velocity and θ0 is the initial temperature. As derived and mentioned in 8 2D g-B enard problem arises in a natural way when we study the standard 3D B enard problem on the thin domain Ωg Ω 0 g . Here the g-B enard problem is a couple system which consists of g-Navier-Stokes equations and the advection-diffusion heat equation in order to model convection in a fluid. Moreover when g const we get the usual B enard problem and when θ 0 we get the g-Navier-Stokes equations. In what follows we will list some related results. Received June 5 2020. Revised June 19 2020. Accepted June 26 2020 Contact Le Thi Thuy e-mail address thuylt@ 23 Tran Quang Thinh and Le Thi .
