Applied Analysis for Nonlinear Systems

(1)热对流问题：为了将局部分岔理论得到的分岔曲线扩展到解空间的解析未知区域，研究解在扩展的分岔曲线上的稳定性变化，并了解全局分岔结构，采用新的计算机辅助分析方法对Boussinesq方程进行了求解。特别地，我们用计算机辅助证明了滚型解的扩展分岔曲线的存在性。给出了在扩展分岔曲线上确定二次分岔点的方法。(2)采用修正的高雷诺数数值验证方法，证明了Navier-Stokes方程存在腔体流动。我们重新表述了无限维空间中不动点方程的牛顿法。(3)证明了Navier-Stokes方程解的爆破是由两个涡度分量来表征的，这意味着它的三个分量不需要保护爆破。(4)用牛顿法研究了无限维Banach空间中的强迫非线性波动方程。线性化方程在近似解（由计算机构造）处的逆算子可以用伪对角算子在范数中逼近。(5)在Lorenz模型方程中证明了奇简并异斜环的存在性，它是一个不变量集。我们假设它会通过扰动给出混沌吸引子。

(1)Heat Convection Pronlem : To extend the bifurcation curves obtained by the local bifurcation theory into the analytically unknown region in the solution space, to investigate the change of stability of the solution on the extended bifurcation curves and to know the global bifurcation structure, we use new computer assisted analysis for Boussinesq equation. Especially, we showed by computer assisted proofs the existence of extended bifurcation curves of the roll-type solutions. We also formulate a method to determine the point of secondary bifurcation on the extended bifurcation curves.(2)The cavity flows of Navier-Stokes equation are proved to exist by a revised numerical verification method for the higher Reynolds number. We reformulated the Newton method for the fixed point equation in the infinite dimensional space.(3)The blow-up of the solution of Navier-Stokes equation is proved to be characterized by the two components of vorticity, which means that its three components are not necessary to protect the blow-up.(4)Forced nonlinear wave equations are investigated by the Newton method in the infinite dimensional Banach space. The inverse operator of linearized equation at the approximate (constructed by computers) solution can be approximated in the norm by a pseudo diagonal operator.(5)In the Lorenz model equation we proved the existence of singularly degenerate heteroclinic cycle, which is an invariant set. We suppose that it will give the chaotic attractor by a perturbation.