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.


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