Degenerate bifurcation of the rotating patches

Taoufik Hmidi, Joan Mateu

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we study the existence of doubly-connected rotating patches for Euler equations when the classical non-degeneracy conditions are not satisfied. We prove the bifurcation of the V-states with two-fold symmetry, however for higher m-fold symmetry with m≥3 the bifurcation does not occur. This answers to a problem left open in [10]. Note that, contrary to the known results for simply-connected and doubly-connected cases where the bifurcation is pitchfork, we show that the degenerate bifurcation is actually transcritical. These results are in agreement with the numerical observations recently discussed in [10]. The proofs stem from the local structure of the quadratic form associated to the reduced bifurcation equation.

Original languageEnglish (US)
Pages (from-to)799-850
Number of pages52
JournalAdvances in Mathematics
Volume302
DOIs
StatePublished - Oct 22 2016

Keywords

  • Degenerate bifurcation
  • Euler equations
  • Rotating patches

ASJC Scopus subject areas

  • Mathematics(all)

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