A nonlinear approximation for vortex sheet evolution and singularity formation

Russel E. Caflisch; Stephen Semmes

doi:10.1016/0167-2789(90)90122-6

A nonlinear approximation for vortex sheet evolution and singularity formation

Russel E. Caflisch, Stephen Semmes

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The evolution of a vortex sheet in two-dimensional, incompressible, inviscid flow is governed by the integro-differential equation of Birkhoff-Rott. We derive a simple approximation for vortex sheet evolution, consisting of a system of four first-order differential equations. This approximate system has the advantage of involving only local operators. The errors in the approximation are shown to be relatively small even if the sheet has infinite curvature at a point. For the approximate equations, exact similarity solutions exhibiting singularity formation are constructed.

Original language	English (US)
Pages (from-to)	197-207
Number of pages	11
Journal	Physica D: Nonlinear Phenomena
Volume	41
Issue number	2
DOIs	https://doi.org/10.1016/0167-2789(90)90122-6
State	Published - Mar 1990

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
Condensed Matter Physics
Applied Mathematics

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10.1016/0167-2789(90)90122-6

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@article{21073acc12ce459f97daed0fe8133d8e,

title = "A nonlinear approximation for vortex sheet evolution and singularity formation",

abstract = "The evolution of a vortex sheet in two-dimensional, incompressible, inviscid flow is governed by the integro-differential equation of Birkhoff-Rott. We derive a simple approximation for vortex sheet evolution, consisting of a system of four first-order differential equations. This approximate system has the advantage of involving only local operators. The errors in the approximation are shown to be relatively small even if the sheet has infinite curvature at a point. For the approximate equations, exact similarity solutions exhibiting singularity formation are constructed.",

author = "Caflisch, {Russel E.} and Stephen Semmes",

note = "Funding Information: 1Research supported in part by the Air Force Office of Scientific Research under U.RI grant 86-0352 and grant 90-0003 and by the National Science Foundation under grant NSF-MCS-83-01260. Current address: Mathematics Department, UCLA, Los Angeles, CA 90024, USA. 2 Research supported by the National Science Foundation and the Alfred P. Sloan Foundation.",

year = "1990",

month = mar,

doi = "10.1016/0167-2789(90)90122-6",

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volume = "41",

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T1 - A nonlinear approximation for vortex sheet evolution and singularity formation

AU - Caflisch, Russel E.

AU - Semmes, Stephen

N1 - Funding Information: 1Research supported in part by the Air Force Office of Scientific Research under U.RI grant 86-0352 and grant 90-0003 and by the National Science Foundation under grant NSF-MCS-83-01260. Current address: Mathematics Department, UCLA, Los Angeles, CA 90024, USA. 2 Research supported by the National Science Foundation and the Alfred P. Sloan Foundation.

PY - 1990/3

Y1 - 1990/3

N2 - The evolution of a vortex sheet in two-dimensional, incompressible, inviscid flow is governed by the integro-differential equation of Birkhoff-Rott. We derive a simple approximation for vortex sheet evolution, consisting of a system of four first-order differential equations. This approximate system has the advantage of involving only local operators. The errors in the approximation are shown to be relatively small even if the sheet has infinite curvature at a point. For the approximate equations, exact similarity solutions exhibiting singularity formation are constructed.

AB - The evolution of a vortex sheet in two-dimensional, incompressible, inviscid flow is governed by the integro-differential equation of Birkhoff-Rott. We derive a simple approximation for vortex sheet evolution, consisting of a system of four first-order differential equations. This approximate system has the advantage of involving only local operators. The errors in the approximation are shown to be relatively small even if the sheet has infinite curvature at a point. For the approximate equations, exact similarity solutions exhibiting singularity formation are constructed.

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A nonlinear approximation for vortex sheet evolution and singularity formation

Abstract

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