The primary research objectives of this project are: to understand the long time dynamics of nearly flat vortex sheets; to determine analytically the stability of solitary waves in bi-directional model systems of vortex sheets; and to study the behavior of interfaces between electrified fluids. The first project is rooted in proving dispersive estimates for the linearized vortex sheet problem along with a priori energy estimates for the nonlinear problem. The ultimate goal is to show that if the initial configuration is nearly flat then solutions of the full nonlinear equations of motion exist globally in time. The second area is focused on understanding and circumventing technical obstructions which arise when trying to establish orbital stability for solitary waves in approximate models for free fluid interfaces derived without the assumption that there is a preferred direction of propagation. Uni-directionality plays an important but subtle role in the stability analysis and generalization to bi-directional systems has proven difficult. The third project is centered around understanding how an applied electric field can exacerbate/arrest rupture of the interface. The electro-magnetics introduce new nonlocal and nonlinear effects which complicate the analysis of the system.
Predicting how an interface between two fluids which are shearing past one another moves in time is a central problem in mathematical hydrodynamics. This very general scenario occurs in many situations of practical interest: on the surface of the ocean, between layers in the atmosphere, in the wake of a boat or behind the wing of an aircraft. A natural question to ask is whether or not an interface initially very close to being perfectly flat will in fact become perfectly flat as time evolves. In the first part of this research project, very precise quantitative descriptions of nearly flat interfaces will be developed. The second component of this project concerns the evolution of solitary water waves, which are called tsunamis when they occur in the open ocean. The destructive power of these waves is directly related to the fact that they are very difficult to disrupt: they can travel great distances essentially unchanged. That solitary waves are so stable is well understood for a variety of models for their evolution wherein one makes an assumption that the wave has a preferred direction of motion. This part of this project studies their stability without this assumption, and also to examine ways in which bottom topography can (possibly) disturb such waves. The final part of this project is concerned with developing and analyzing systems which model the effects of an applied electromagnetic field on a fluid interface. There is a great deal of interest in manipulated fluid interfaces in this way. Applications include: dynamic wave guides and lenses, high speed switching, coating and cooling processes.
