Applied Analysis for Nonlinear Systems

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(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.