Toward a Global Analysis for Nonlinear System of Partial Differential Equations

研究了可压缩、粘性和导热流体的对流换热问题。证明了当Rayleigh数超过临界值时，从平衡态(热传导解)开始发生定常分叉。它们对应于滚筒型溶液、六角晶胞和混合型溶液的花样形成。该分叉关于对应于温度梯度的参数均匀地发生，并且当该参数趋于零时收敛于Oberbeck-Boussinesq方程的分叉。采用Lax-Friedrichs格式构造时间周期解，数值得到周期解、混沌解和Aubry-Mather集。

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.