Toward a Global Analysis for Nonlinear System of Partial Differential Equations

研究了可压缩、粘性、导热流体的对流换热问题。稳态分岔被证明发生从平衡状态（热传导解决方案），当瑞利数超过临界值。它们对应于卷型溶液、六角胞和混合型溶液的图案形成。当温度梯度参数趋于零时，分岔点一致收敛于Oberbeck-Boussinesq方程的分岔点。研究了汉密尔顿力学的Jauslin-Kreiss-Moser模型。这是一个具有周期边界条件的周期受迫Burgers方程，利用Lax-Friedrichs方法构造了该方程的周期时间解，并数值计算了该方程的周期解、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.