Attractors for two-dimensional equations of thermal convection in the presence of the dissipation function.(English)Zbl 0843.35074

The author studies the Bénard convection problem with dissipative heating in a domain $$\Omega= (0, \alpha)\times (0, 1)\subset \mathbb{R}^2$$, i.e. $u_t- \nu\Delta u+ (u\cdot \nabla) u+ \nabla p= e_2 f(\theta),\quad\text{div } u= 0\quad\text{in } \Omega\times (0, \infty),$
$\theta_t- \kappa\Delta\theta+ u\cdot \nabla\theta- e_2\cdot u= {\eta\nu\over 2} D(u): D(u)\quad\text{in } \Omega\times (0, \infty),$ together with the boundary conditions $$u$$, $$\theta= 0$$ on $$x_2= 0, 1$$ while the solution is required to be periodic in $$x_1$$. The dissipation function is given by $D(u)= \sum^2_{i,k= 1} \Biggl({\partial u^i\over \partial x_k}+ {\partial u^k\over \partial x_i}\Biggr)^2$ and $$f$$ is a smooth function satisfying either $$f(\theta)= \theta$$ or $$\sup|f, f', f''|< \infty$$.
Using energy estimates, the author establishes existence and uniqueness of solutions as well as their continuous dependence on the initial data. After that he proves the existence of a global attractor and gives an estimate for its Hausdorff dimension involving the relevant physical parameters.

MSC:

 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 76R10 Free convection