Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion

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

“…( In [7] w e proved that under the assumptions then the solution is unique. In this paper we shall show that, under suitable assumptions on the data, spatial or temporal localization of the temperature component o f a n y w eak solution occurs.…”

“…In [7] w e studied the well posedness of the system (1.2) together with suitable initial and boundary conditions under suitable assumptions on the data including the cases of degenerate and singular parabolicity of the heat equation. This enforced, for well known reasons, the introduction of a notion of weak solution for the heat equation.…”

On Stability of Solutions to Equations Describing Incompressible Heat-Conducting Motions Under Navier’s Boundary Conditions

“…Papers [6,8,10,16,19] are concerned with solvability results for the Boussinesq equations, that is, for equations (1) 1,2,3 , where ν|D(v)| 2 disappears. Kagei in [10] and Morimoto in [16] proved the existence of weak solutions to the three-dimensional Boussinesq equations with temperature -independent viscosity, while in [8] the weak solvability of the three-dimensional problem with temperature -dependent viscosity has been examined.…”

“…Kagei in [10] and Morimoto in [16] proved the existence of weak solutions to the three-dimensional Boussinesq equations with temperature -independent viscosity, while in [8] the weak solvability of the three-dimensional problem with temperature -dependent viscosity has been examined. Diaz and Galiano [6] showed existence of weak solutions to the following initialboundary value problem…”

“…In paper [22] we proved the existence of a strong global solution to (1) 1,2,3,6,7 with the slip boundary conditions for velocity which are close for all time to a two-dimensional solution.…”