Dynamics near unstable, interfacial fluids

We study three examples of unstable interfacial fluid motions: vortex sheets with surface tension, Hele-Shaw flows with surface tension, and vortex patches. In all three cases, the nonlinear dynamics of a large class of smooth perturbations is proven to be characterized by the corresponding fastest linear growing mode(s) up to the time scale of log 1\δ }, where δ is the magnitude of the initial perturbation. In all three cases, the analysis is based on an unified analytical framework that includes precise bounds on the growth of the linearized operator, given by an explicit solution formula, as well as a special sharp nonlinear energy growth estimate. Our main contribution is establishing this nonlinear energy growth estimate for each interface problem in certain high energy norms.