Two-dimensional euler flows with concentrated vorticities

Manuel Del Pino, Pierpaolo Esposito, Monica Musso

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27 Citations (SciVal)


For a planar model of Euler flows proposed by Tur and Yanovsky (2004), we construct a family of velocity fields ws for a fluid in a bounded region Q, with concentrated vorticities wε for ε ≥ 0 small. More precisely, given α positive integer a and a sufficiently small complex number a, we find a family of stream functions ψ ε which solve the Liouville equation with Dirac mass source, Δ ψε + ε2εψε =4π αδpaε in Ω,ψε = 0 on ω, for a suitable point p = pa,εε ω. The vorticities Wε = - Δφε concentrate in the sense that [Eqation Present] where the satellites ai,⋯,aα+1 denote the complex (a + 1)-roots of a.The point pa<s lies close to a zero point of a vector field explicitly built upon derivatives of order ≤ α + 1 of the regular part of Green's function of the domain.

Original languageEnglish
Pages (from-to)6381-6395
Number of pages15
JournalTransactions of the American Mathematical Society
Issue number12
Publication statusPublished - 1 Dec 2010


  • 2D Euler equations
  • Concentrating solutions
  • Liouville formula
  • Singular Liouville equation

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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