Degenerate bifurcation of the rotating patches

Taoufik Hmidi (1) , Joan Mateu (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 UAB - Departament de Matemàtiques [Barcelona]

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 twofold symmetry, however for higher m−fold symmetry with m ≥ 3 the bifurcation does not occur. This answers to a problem left open in [19]. 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 [19]. The proofs stem from the local structure of the quadratic form associated to the reduced bifurcation equation.

Domaines: Equations aux dérivées partielles [math.AP]

Taoufik Hmidi, Joan Mateu. Degenerate bifurcation of the rotating patches. Advances in Mathematics, 2016, 302, pp.799-850. ⟨10.1016/j.aim.2016.07.022⟩. ⟨hal-01365280⟩