The proposed research concerns asymptotics of conservative partial differential equations (PDE) and instability of the incompressible Euler equations in three specific areas. The first project concerns the long time behavior of vortex solutions in the nonlinear wave equation. Although recent progress has led to greater understanding of how concentrations behave in the nonlinear wave equation, very little is known about how such solutions radiate energy after long times; we will rigorously study this question. Second, we will study the nonlinear wave equation in asymptotically thin domains. Conventional wisdom holds that most PDE's with an asymptotically large aspect ratio will be well approximated simply by dropping the dimension of the PDE and relying only on the planar direction; this work will examine the precise timescales for which this practice is acceptable. The third project involves the nonlinear instability of various PDE's arising from the incompressible Euler equations, in which the governing dynamics reduce to the behavior of an interface -- such as in two-layer models and vortex patches.
Often, physical phenomena are well modeled via nonlinear PDE's. Such equations are exceedingly difficult to study, both numerically and theoretically, yet their understanding is crucial to the further progress of many areas of physics and engineering. Two of the projects outlined above offer rigorous methods for simplifying classes of these equations, not only providing insight into the basic behavior of fluids and quantum field theory, but also greatly reducing the scope of numerical simulations. Such reductions in computational costs should aid physicists, computer scientists, and engineers. The third project endeavors to classify the stable structures in fluid dynamics, helping us to gain a better qualitative understanding of the way fluids behave.
