Abstract. We prove the existence of the V-states for the generalized inviscid SQG equations with α ∈]0, 1[. These structures are special rotating simply connected patches with m− fold symmetry bifurcating from the trivial solution at some explicit values of the angular velocity. This produces, inter alia, an infinite family of non stationary global solutions with uniqueness.

We rigorously construct continuous curves of rotating vortex patch solutions to the twodimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum along the interface of the angular fluid velocity in the rotating frame becomes arbitrarily small. This is consistent with the conjectured existence [WOZ84, Ove86] of singular limiting patches with 90 • corners at which the relative fluid velocity vanishes. For solutions close to the disk, we prove that there are "Cat's eyes"-type structures in the flow, and provide numerical evidence that these structures persist along the entire solution curves and are related to the formation of corners. We also show, for any rotating vortex patch, that the boundary is analytic as soon as it is sufficiently regular.

26 pagesWe deal with the local well-posedness theory for the two-dimensional inviscid Boussinesq system with rough initial data of Yudovich type. The problem is in some sense critical due to some terms involving Riesz transforms in the vorticity-density formulation. We give a positive answer for a special sub-class of Yudovich data including smooth and singular vortex patches. For the latter case we assume in addition that the initial density is constant around the singular part of the patch boundary