Existence and dimension of the attractor for the Bénard problem on channel-like domains.
The Bénard system is a system with the Navier-Stokes equations for the velocity field coupled with a convection-diffusion equation for the temperature. The velocity influences the diffusion of the temperature and the temperature induces a convective motion of the fluid via the Boussinesq approximation. For this problem, the authors prove the existence of global solutions and of the global attractor in the usual \(L^2\)-like space. Also, it is proved that the global attractor has finite Hausdorff and fractal dimensions, which are estimated in terms of suitable Reynolds, Prandtl, Grashof and Rayleigh numbers. The used approach, unlike the previous works, avoids the use of the maximum principle which would be problematic in this context.
scientific article; zbMATH DE number 2077709