Attractors for the Bénard problem: existence and physical bounds on their fractal dimension

Structure of bifurcated solutions of two-dimensional infinite Prandtl number convection with no-slip boundary conditions

“…By [7], we know that the solution (u, T ) is analytic in time and all H m norms of (u(t, Ψ 0 ), T (t, Ψ 0 )) remains uniformly bounded in time for t ≥ δ > 0. Therefore by Theorem 3.5, we have where K = min{k | α k = 0, k is odd}, and t is sufficiently large.…”

“…In particular, for the RayleighBénard convection with the finite Prandtl numbers, readers are referred to Chandrasekhar [2], to Drazin and Reid [5] for linear theories; to Foias, Manley and Temam [7] for the existence and physical bounds of attractors; to Rabinowitz [18] for the existence of rectangular solutions; and to Ma and Wang [12] for attractor bifurcation. For the case regarding the infinite Prandtl number, the readers are referred to Constantin and Doering [3,4] for the upper bounds of the minimal conduction value; to Schnaubelt and Busse [21] for two-dimensional convection rolls; to Keken [9] and to Yanagisawa and Tamagishi [25] for mantle and spherical shell convection studies, respectively; to X. Wang [23,24] for the justification of the infinite Prandtl number convection as the infinite Prandtl number limit of the Boussinesq equations; and to Park [15,16] for the attractor bifurcation theory of the infinite Prandtl number convection.…”

Two-Dimensional Infinite Prandtl Number Convection: Structure of Bifurcated Solutions

“…Proof. By [6], we know that the solution (u(t, ψ 0 ), T (t, Ψ 0 )) is analytic in time and all the H m norms of u(t, ψ 0 ), T (t, Ψ 0 ) remain uniformly bounded in time for t ≥ δ > 0, m ≥ 0.…”

“…In particular, for the Rayleigh-Bénard convection with the finite Prandtl numbers, readers are referred to Chandrasekhar [1], to Drazin and Reid [4] for linear theories; to Foias, Manley and Temam [6] for the existence and physical bounds of attractors; to Rabinowitz [16] for the existence of rectangular solutions; and to Ma and Wang [10,11] for attractor bifurcation. For the case regarding the infinite Prandtl number, the readers are referred to Constantin and Doering [2,3] for the upper bounds of the minimal conduction value; to Schnaubelt and Busse [19] for two-dimensional convection rolls; to Keken [8] and to Yanagisawa and Tamagishi [23] for mantle and spherical shell convection studies, respectively; to X. Wang [21,22] for the justification of the infinite Prandtl number convection as the infinite Prandtl number limit of the Boussinesq equations; and to Park [14] for the existence of attractor bifurcation.…”

Spatial and time localization of solutions of the Boussinesq system with nonlinear thermal diffusion

“…Density di erences are induced, for instance, by gradients of temperature arising by non uniform heating of the uid. In the Boussinesq approximation of a large class of ow problems thermodynamical coe cients such as viscosity, speci c heat and thermal conductivity, can be assumed constant leading to a coupled system with linear second order operators in the Navier-Stokes and heat equations (see, e.g., [10], [11], [15], [21]). However, there are some uids like lubri cants or some plasma ow for which this is no longer an accurate assumption (see, e.g., [14], [19]).…”