The point-vortex method for periodic weak solutions of the 2-D euler equations

Steven H. Schochet*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 Lp 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.

Original languageEnglish
Pages (from-to)911-965
Number of pages55
JournalCommunications on Pure and Applied Mathematics
Issue number9
StatePublished - Sep 1996


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