TY - JOUR

T1 - Long time existence for a slightly perturbed vortex sheet

AU - Caflisch, Russel E.

AU - Orellana, Oscar F.

PY - 1986/11

Y1 - 1986/11

N2 - 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 |풥m 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 tc = |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.

AB - 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 |풥m 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 tc = |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.

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U2 - 10.1002/cpa.3160390605

DO - 10.1002/cpa.3160390605

M3 - Article

AN - SCOPUS:84990556256

SN - 0010-3640

VL - 39

SP - 807

EP - 838

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

IS - 6

ER -