Existence theory and strong attractors for the Rayleigh-Bénard problem with a large aspect ratio

Abstract

Abstract. The Navier-Stokes equation driven by heat conduction is studied. It is proven that if the driving force is small then the solutions of the Navier-Stokes equation are ultimately regular. As a prototype we consider Rayleigh-Bénard convection, in the Boussinesq approximation. Under a large aspect ratio assumption, which is the case in Rayleigh-Bénard experiments with Prandtl numer close to one, we prove the ultimate existence and regularity of a global strong solution to the 3D Navier-Stokes equation coupled with a heat equation, and the existence of a maximal B-attractor. Examples of simple B-attractors from pattern formation are given and a method to study their instabilities proposed.1. Introduction. In this paper we study the Navier-Stokes equation driven by heat conduction. The existence of strong solutions to the Navier-Stokes equation is still open in three dimensions. On the other hand it has long been known that weak (L 2 ) solutions exist, [32], in three dimensions and if the initial data and the forcing is small then strong solutions exist, see [33,34] [52]. In this paper we show that it suffices to assume the latter of the above conditions, namely the smallness of the forcing, to get ultimate regularity of solutions and the existence of global attractors in three dimensions consisting of smooth solutions. The proof of this results which is stated as Theorem 2.2 is very simple. The existence of an attracting set for weak solutions of the three-dimensional Navier-Stokes equations was proven by Foias and Temam,[20], and Sell [50] proved the existence of a global attractor of weak solutions, without assuming that the forcing is small. Thus our hypothesis is stronger than Sell's but our result is also stronger because it is not clear whether his solutions are ultimately smooth, whereas our attractor consists of strong solutions and has all the nice properties that follow. Many authors have proven the existence of strong solutions and attractors in thin three-dimensional domains, see Raugel [47] and the references therein, and [46,27], etc. None of these results apply to our study of the Rayleigh-Bénard problem. The reason is that we will not assume that the domain is thin unless the driving is very large and even in the latter case the results in thin domains do not apply because we do not restrict the initial data. Instead we will

The Oberbeck–Boussinesq approximation and Rayleigh–Bénard convection revisited

“…Wang [16] consider the same problem in the large Prandtl number regime, where the fluid velocity becomes regular for long times. Similar ideas were exploited by Birnir and Svanstedt [4]. Notice that here ∇G could also include centrifugal force (cf., e.g., [8]).…”

Regularity and attractors for the three‐dimensional generalized Boussinesq system

“…In the mathematical respect, the global well‐posedness, global regularity of the 3D Boussinesq system, and the existence of the global attractor have been widely studied (see previous studies [6, 8, 11–24] and references therein) and the 3D generalized Boussinesq equations has caught much attention recently (see, e.g., previous studies [25–34]). This family of the 3D generalized Boussinesq equations can be considered as an interpolation model between subcritical, critical, and supercritical dissipations as varying the parameters: In case of $$$ \alpha =\beta =1 $$$, () is known as the 3D Boussinesq system (also known as the Bénard problem) (see, e.g., Kapustyan et al [15]); in case of $$$ \alpha =\beta =0 $$$, we get the damped 3D Boussinesq equations without diffusion (see, e.g., Li et al [19]), and in addition, if $$$ \theta =0 $$$, () becomes the 3D damped Euler equations (see, e.g., Gibbon [35] and Ilyin et al [36]); in case of $$$ \alpha \ge \frac{5}{4},\beta =1 $$$, () reduces to the hyperdissipative Boussinesq equations (see, e.g., Yang et al [33]); and in case of $$$ \alpha >0,\kappa =0 $$$, we receive the 3D generalized Boussinesq equations without thermal diffusion (see, e.g., previous studies [26, 28, 34]).…”

“…As far as we know, the results for the 3D generalized Boussinesq equations are restricted and let us recall them here. In Birnir and Svanstedt [6], the global attractor for the 3D Bénard problem in the natural phase space is constructed only for “small” external forces. Kapustyan et al [15] study the asymptotic behavior of weak solutions for the 3D Bénard problem.…”

“…The Boussinesq system is a well-known model in hydrodynamics and describes the behavior of the velocity, the pressure, and the temperature of an incompressible fluid. A detailed description of the physical background can be found in Birnir and Svanstedt [6] and Temam [7]. For this reason, this system is studied systematically by scientists from different domains; see, for instance, previous studies [7][8][9][10].…”

Global attractor for the time discretized modified three-dimensional Bénard systems

“…The discretization of differential equation mainly refers to the discretization of time and space. The usual discretization methods include finite difference method, finite element method, weighted residual method and so on (see [1,2,4,6,7,14,16]). In [5], the modified 3D Navier-Stokes equations were discretized on the time by finite difference method, then the existence of the global attractor was proved.…”