Existence and nonlinear stability of convective solutions for almost compressible fluids in Bénard problem

Abstract

We study the nonlinear almost compressible 2D Oberbeck-Boussinesq system, characterized by an extra buoyancy term where the density depends on the pressure, and a corresponding dimensionless parameter β, proportional to the (positive) compressibility factor β 0. The local in time existence of the perturbation to the conductive solution is proved for any "size" of the initial data. However, unlike the classical problem where β 0 = 0, a smallness condition on the initial data is needed for global in time existence, along with smallness of the Rayleigh number. Removing this condition appears quite challenging and we leave it as an open question.

Existence and Uniqueness of Isothermal, Slightly Compressible Stratified Flow

“…Let us denote by H −1 the dual space of H. We show the following lemma that in the case α < 2π was proved in [2,4] by different arguments. Since F ∈ L 2 (Ω 0 ), by classical elliptic regularity we get that q ∈ W 2,2 (Ω 0 ), and that it satisfies (4.5) along with q 2,2 ≤ C M 2 .…”

“…Motivated by this important issues, the author jointly with D. Grandi rigorously derived, by perturbative methods from the full set of balance laws, new models where the density, ρ, may depend on both T and p [12,13], while for related ones, the authors addressed well-posedness and stability questions, in [2,4,5]. In particular, in [13], we proposed a new model for thermal convection in a horizontal layer of fluid heated from below with ρ = ρ(T, p).…”

Bénard Problem for Slightly Compressible Fluids: Existence and Nonlinear Stability in 3D

“…As is well known, one of the most accepted models in the study of convection problems in fluid mechanics is the socalled Oberbeck-Boussinesq approximation (hereafter denoted by O-B) [1][2][3][4][5]. e latter is characterized by the fact that one keeps the fluid incompressible and allows for density changes only in the buoyancy term of the linear momentum equation, by assuming a linear dependence on temperature.…”

“…Notice that (1) tends to the motionless solution for the same problem in the O-B approximation by letting β ⟶ 0. In [4,9,12], the linear and nonlinear stability of r 0 , as well as the existence and uniqueness for the associated "perturbed problem," was investigated. All this work was done in the case of the plane flow.…”

“…e plan of the paper is given in the following. After formulating the problem and collecting some preliminary results in Section 1, in Section 2, the classical Galerkin method is used (see, for instance, [4]) to construct an "approximate" solution. In Section 3, which constitutes the main core of this paper, a number of fundamental estimates on the approximate solution is proven that, again by classical techniques, allows us to conclude that the latter converges to a solution of the original problem having the properties described above.…”