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Long time existence for a slightly perturbed vortex sheet
Author(s) -
Caflisch Russel E.,
Orellana Oscar F.
Publication year - 1986
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160390605
Subject(s) - mathematics , singularity , perturbation (astronomy) , vortex , mathematical analysis , amplitude , vortex sheet , differential equation , physics , vorticity , mechanics , quantum mechanics
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 | 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.