Long time existence and singularity formation for vortex sheets

Abstract

The initial linear evolution of a nearly flat and uniform vortex sheet is given by the Kelvin-Helmholtz instability. Asymptotic analysis and numerical computations of the subsequent nonlinear evolution show several interesting features. At some finite time the vortex sheet develops a singularity in its shape; i.e. the curvature becomes infinite at a point. This is immediately followed by roll-up of the sheet into an infinite spiral. This paper presents two mathematical results on nonlinear vortex sheet evolution and singularity formation: First, for sufficiently small analytic perturbations of the flat sheet, existence of smooth solutions of the Birkhoff-Rott equation is proved almost up to the expected time of singularity formation. Second, we present a construction of exact solutions that develop singularities (infinite curvature) in finite time starting from analytic initial data. These results are derived within the framework of analytic function theory. The analysis of singular solutions is an independent construction of solutions first found by Duchon and Robert (1986,1988). All of these results are in the analytic function setting, since that is the only space in which the vortex sheet problem is known to be well-posed. We present a simple example to show ill-posedness of the 2D Euler equations in the energy norm.KeywordsSingularity FormationEnergy NormPoint VortexVortex SheetSmooth Initial DataThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.