| In this thesis, we study the dynamic transitions of the Rayleigh-Benard convection on both spherical shell and cylindrical domains and the Swift Hohenberg Equation within the framework of dynamic transition theory developed in Ma and Wang.;The Rayleigh-Benard convection of fluids is modeled by the Boussinesq equations. First, we analyze the dynamic transition and pattern formation of the Boussinesq equations on spherical shell, and we show that the system always undergoes a Type-I (continuous) dynamic transition to a 2lc-dimensional homological sphere Sigma R. Moreover, we show that the 2lc dimensional homological sphere SigmaR is in fact homeomorphic to a 2lc-dimensional sphere S2lc. Then we study the phase transition and pattern formation of the Boussinesq equations on cylindrical domain, and obtain very similar results as for the spherical shell domain. Finally, we study the Rayleigh-Benard convection on a spherical shell with infinite Prandtl number. In this case, we show that the dynamic transition problem is governed only by the temperature function T, and the velocity field will depend on T.;For the Swift Hohenberg Equation, we find the reduced equation with two special cases and study phase transition and pattern formation, and also derive a general formula for the reduced equations. |
