### Abstract

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 w_{ε} 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 ψ_{ε} + ε^{2}_{εψε} =4π αδ_{pa}ε in Ω,ψ_{ε} = 0 on ω, for a suitable point p = p_{a},_{ε}ε ω. The vorticities W_{ε} = - Δφ_{ε} concentrate in the sense that [Eqation Present] where the satellites ai,⋯,a_{α+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 language | English |
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Pages (from-to) | 6381-6395 |

Number of pages | 15 |

Journal | Transactions of the American Mathematical Society |

Volume | 362 |

Issue number | 12 |

DOIs | |

Publication status | Published - 1 Dec 2010 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

## Cite this

*Transactions of the American Mathematical Society*,

*362*(12), 6381-6395. https://doi.org/10.1090/S0002-9947-2010-04983-9