### 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 |ð�’¥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.

Original language | English (US) |
---|---|

Pages (from-to) | 807-838 |

Number of pages | 32 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 39 |

Issue number | 6 |

DOIs | |

State | Published - 1986 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*39*(6), 807-838. https://doi.org/10.1002/cpa.3160390605

**Long time existence for a slightly perturbed vortex sheet.** / Caflisch, Russel; Orellana, Oscar F.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 39, no. 6, pp. 807-838. https://doi.org/10.1002/cpa.3160390605

}

TY - JOUR

T1 - Long time existence for a slightly perturbed vortex sheet

AU - Caflisch, Russel

AU - Orellana, Oscar F.

PY - 1986

Y1 - 1986

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 |ð�’¥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 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 |ð�’¥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 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.

UR - http://www.scopus.com/inward/record.url?scp=84990556256&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84990556256&partnerID=8YFLogxK

U2 - 10.1002/cpa.3160390605

DO - 10.1002/cpa.3160390605

M3 - Article

AN - SCOPUS:84990556256

VL - 39

SP - 807

EP - 838

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 6

ER -