| In this dissertation, I study instability and stability of steady solutions to the one-dimensional Vlasov-Poisson system (VP) and two-dimensional Euler equation of incompressible inviscid fluids (E). Some new methods are developed to find exponentially growing solutions (growing modes) to the linearized equation around the equilibrium. In order to obtain growing modes, I use several different methods, including the concept of neutral modes and of infinite determinant as well as continuation arguments. These methods are used to analyze a corresponding dispersion operator or the neutral mode equations (such as Rayleigh's classical equation). The new results obtained include: the proof of linear instability of any BGK waves of (VP) under multipy-periodic perturbations, new sufficient conditions for instability of plane shear flows and rotating flows for (E) and a criterion of finding purely growing modes for general steady ideal plane flows (E). These methods are flexible and might be applicable to study instability problems of other conservative systems in continuum physics. I also get some stability criteria for quite general ideal plane flows and compare the different types of stability in this case. |
