Long time existence for a slightly perturbed vortex sheet

Abstract

Abstract Consider a flat two‐dimensional vortex sheet perturbed initially by a small analytic disturbance. By a formal perturbation analysis, Moore derived an approximate differential equation for the evolution of the vortex sheet. We present a simplified derivation of Moore's approximate equation and analyze errors in the approximation. The result is used to prove existence of smooth solutions for long time. If the initial perturbation is of size ϵ and is analytic in a strip |𝒥 m γ| < ρ, existence of a smooth solution of Birkhoff's equation is shown for time t < k2p , if ϵ is sufficiently small, with κ → 1 as ϵ → 0. For the particular case of sinusoidal data of wave length π and amplitude e , Moore's analysis and independent numerical results show singularity development at time t c = |log ϵ| + O (log|log ϵ|. Our results prove existence for t < κ|log ϵ|, if ϵ is sufficiently small, with k κ → 1 as ϵ → 0. Thus our existence results are nearly optimal.