Toward a Global Analysis for Nonlinear System of Partial Differential Equations

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Heat convection problems of compressible, viscous and heat-conductive fluids are investigated. Stationary bifurcations are proved to occur from the equilibrium state ( heat conduction solution ), when the Rayleigh number exceeds the critical value. They correspond to the pattern formation of the roll-type solution, the hexagonal cell and the mixed type solution. The bifurcation occurs uniformly with respect to the parameter which corresponds to the gradient of temperature and they converge to those of the Oberbeck-Boussinesq equation as the parameter tends to zero.Jauslin-Kreiss-Moser model of Hamilton Mechanics is investigated. It is a periodically forced Burgers equation with the periodic boundary condition.The periodic in time solutions are constructed by the Lax-Friedrichs scheme and the periodic, chaotic solutions and the Aubry-Mather set are obtained numerically.