Applied Analysis for Nonlinear Systems

（1）热对流问题：为了将局部分歧理论得到的分歧曲线推广到解空间中的解析未知区域，研究解在推广的分歧曲线上的稳定性变化，了解全局分歧结构，我们对Boussinesq方程进行了新的计算机辅助分析。特别地，通过计算机辅助证明证明了该滚动型解的扩张分支曲线的存在性。我们还制定了一个方法来确定扩展的分歧曲线上的二次分歧点。（2）采用修正的数值验证方法，证明了高雷诺数下Navier-Stokes方程中空腔流动的存在性。我们将牛顿法推广到无限维空间中的不动点方程。（3）证明了Navier-Stokes方程解的爆破是由涡量的两个分量来刻画的，这意味着它的三个分量并不是保证爆破的必要条件。（4）利用Newton方法研究了无穷维Banach空间中的强迫非线性波动方程。线性化方程的逆算子在近似解（由计算机构造）处可用一个伪对角算子在范数下近似。(5)In证明了Lorenz模型方程的奇异退化异宿环的存在性，它是一个不变集。我们假设它会通过一个扰动产生混沌吸引子。

(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.