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The point‐vortex method for periodic weak solutions of the 2‐D Euler equations
Author(s) -
Schochet Steven
Publication year - 1996
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/(sici)1097-0312(199609)49:9<911::aid-cpa2>3.0.co;2-a
Subject(s) - mathematics , euler equations , vortex , euler's formula , mathematical analysis , point (geometry) , euler method , geometry , mechanics , physics
Almost‐sure convergence of a subsequence of the vorticity to a weak solution is proven for the point‐vortex method for 2‐D, inviscid, incompressible fluid flow. Here “almost‐sure” is with respect to sequences of random components included in the initial position and strength of each vortex. The initial vorticity is assumed to be periodic and, depending on the initialization scheme, to lie in L log L or L p with p > 2. The randomization of the initial data is not needed when the initial vorticity is nonnegative; such initial data also need not be periodic, and is only required to be a bounded measure lying in H −1 . All these results are also valid for the “vortex‐blob” method with the smoothing parameter vanishing at an arbitrary rate. The sense in which solutions of point‐vortex dynamics are weak solutions of the Euler equations is also discussed.