The paper deals with singular first order Hamiltonian systems of the formwhere J ∈ R 2×2 defines the standard symplectic structure in R 2 , and the Hamiltonian H is of N -vortex type:This is defined on the configuration space {(z 1 , . . . , z N ) ∈ Ω 2N : z j = z k for j = k} of N different points in the domain Ω ⊂ R 2 . The function F : Ω N → R may have additional singularities near the boundary of Ω N . We prove the existence of a global continuum of periodic solutions z(t) = (z 1 (t), . . . , z N (t)) ∈ Ω N that emanates, after introducing a suitable singular limit scaling, from a relative equilibrium Z(t) ∈ R 2N of the N -vortex problem in the whole plane (where F = 0). Examples for Z include Thomson's vortex configurations, or equilateral triangle solutions. The domain Ω need not be simply connected. A special feature is that the associated action integral is not defined on an open subset of the space of 2π-periodic H 1/2 functions, the natural form domain for first order Hamiltonian systems. This is a consequence of the singular character of the Hamiltonian. Our main tool in the proof is a degree for S 1 -equivariant gradient maps that we adapt to this class of potential operators.MSC 2010: Primary: 37J45; Secondary: 37N10, 76B47

In this article we consider the inhomogeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity as a differential inclusion. By considering the relaxation of the constitutive laws we formulate a general criterion for the existence of infinitely many weak solutions which reflect the turbulent mixing of the two fluids. Our criterion can be verified in the case that initially the fluids are at rest and separated by a flat interface with the heavier one being above the lighter one—the classical configuration giving rise to the Rayleigh–Taylor instability. We construct specific examples when the Atwood number is in the ultra high range, for which the zone in which the mixing occurs grows quadratically in time.

We examine the N -vortex problem on general domains Ω ⊂ R 2 concerning the existence of nonstationary collision-free periodic solutions. The problem in question is a first order Hamiltonian system of the formwith some regular and symmetric, but in general not explicitely known function g : Ω × Ω → R. The investigation relies on the idea to superpose a stationary solution of a system of less than N vortices and several clusters of vortices that are close to rigidly rotating configurations of the whole-plane system. We establish general conditions on both, the stationary solution and the configurations, under which multiple T -periodic solutions are shown to exist for every T > 0 small enough. The crucial condition holds in generic bounded domains and is explicitely verified for an example in the unit disc Ω = B 1 (0). In particular we therefore obtain various examples of periodic solutions in B 1 (0) that are not rigidly rotating configurations.MSC 2010: Primary: 37J45; Secondary: 37N10, 76B47

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