| This thesis consists of three parts. First one examines the problem of stability of stratified flows. This type of flows occur all the time in nature, with the atmosphere and ocean as the two prime examples. We derive the equations for stratified flows in hydrostatic balance in both multilayer and continuous formulations. We introduce a novel stability criterion for stratified flows, which re-interprets stability in terms not of growth of small perturbations, but of the local well-posedness of the time evolution. This reinterpretation allows one to extend the classic results of Miles and Howard concerning steady and planar flows, to the realm of flows that are non-uniform and unsteady.;The second part of this thesis involves the study of simple waves in phase space. They are fully nonlinear solutions to quasi-linear systems of PDEs. In phase space, where they are represented as solutions to nonlinear ODEs and converge to points of crossing eigenvalues of the matrix of the original quasi-linear system. This peculiar phenomenon contradicts the result of Wigner and von Neumann, who discovered in 1929 that eigenvalues "avoidance of crossing".;In the third part, we return to the study of nonlinear stability of systems of conservation laws via simple waves. We show that for two-dimensional systems, simple waves are natural stability bounds for solutions of quasi-linear PDEs -- the regions bounded away from the elliptic domain by simple waves are nonlinearly stable -- any solution starting in them will remain hyperbolic until the breaking time. The planned future work is to extend this result to higher dimensions. |
