Attractors for autonomous double‐diffusive convection systems based on Brinkman–Forchheimer equations

Abstract

Communicated by M. EfendievIn this paper, we consider the existence of global attractor and exponential attractor for some dynamical system generated by nonlinear parabolic equations in bounded domains with the dimension N Ä 4 which describe double-diffusive convection phenomena in a porous medium. We deal with both of homogeneous Dirichlet and Neumann boundary condition cases. Especially, when Neumann condition is imposed, we need some assumptions and restrictions for the external forces and the average of initial data, since the mass conservation law holds.

Global solvability of some double-diffusive convection system coupled with Brinkman-Forchheimer equations

“…Moreover, when the constant vectors g, h are very large, it is difficult to examine whether condition (A6) is satisfied. From these reasons, we are led to introduce the same type of relaxed approximate problems as in [11].…”

“…In [11], the global solvability of the time periodic problem is shown for (BF) with the homogeneous Dirichlet boundary condition both for 2 and 3-dimensional cases.…”

“…Since (BF) contains the convection terms u · ∇ T, u · ∇ C, which are quite similar to that appearing in the Navier-Stokes equations, it apparently seems that it would be very difficult to obtain " the global solvability " of (BF) in 3-dimensional case, i.e., the existence of the unique global solution of the initial boundary-value problem for arbitrarily large initial data or the existence of time-periodic solutions for arbitrarily large external forces. However, it is revealed that the global solvability holds true for these problems even for the 3-dimensional case in [12] and [11].…”

“…The main purpose of this paper is to show the similar global solvability results still hold true for (BF) with the homogeneous Neumann boundary condition for T and C. In order to carry out this purpose, we follow the basic strategy adopted in [12] and [11], i.e., we reduce our problem to some abstract equation in an appropriate Hilbert space and we rely on the abstract theory developed in [9] and [10]. However, the lack of the coercivity of the Laplacian with the Neumann boundary condition causes some difficulty in this procedure.…”

“…However, the lack of the coercivity of the Laplacian with the Neumann boundary condition causes some difficulty in this procedure. Especially for the periodic problem, we need to introduce some approximate system involving some dissipation terms and cut-off functions as in [11]. Unfortunately this hinders establishing desirable a priori estimates under the Neumann boundary condition.…”