Bifurcation and Stability Problems in Fluid Dynamics

Abstract

My dissertation is to study the bifurcation, stability and phase transitions of incompressible fluid flows. Bifurcation is a versatile methodology to trace solutions of physical problems along with the system parameter and to investigate their structure. The study is oriented toward a nonlinear dynamic theory for the underlying physical problems consisting of 1)complete bifurcation when the system parameter crosses some critical values, 2)asymptotic stability of bifurcated solutions and 3)the structure/pattern of the bifurcated solutions and phase transitions in the physical spaces. The study in the first two directions is related to application of a new bifurcation theory, called attractor bifurcation, which was developed by T. Ma and S. Wang. The third direction of the study is related to geometric study of fluid flows and includes structural stability theory.