  In this article we examine the interaction of incompressible 2D flows with
compact material boundaries. Our focus is the dynamic behavior of the
circulation of velocity around boundary components and the possible exchange
between flow vorticity and boundary circulation in flows with vortex sheet
initial data We begin by showing that the velocity can be uniquely
reconstructed from the vorticity and boundary component circulations, which
allows to recast 2D Euler evolution using vorticity and the circulations as
dynamic variables. The weak form of this vortex dynamics formulation of the
equations is called the weak vorticity formulation. The main result in this
article is the equivalence between the weak velocity and weak vorticity
formulations, without sign assumptions. Next, we focus on weak solutions
obtained by mollifying initial data and passing to the limit, with the portion
of vorticity singular with respect to the Lebesgue measure assumed to be
nonnegative. For these solutions we prove that the circulations around each
boundary component cannot be smaller than the initial data circulation, so that
nonnegative vorticity may be absorbed by the boundary, but not produced by the
boundary. In addition, we prove that if the weak solution conserves circulation
at the boundary components it is a boundary coupled weak solution, a stronger
version of the weak vorticity formulation. We prove existence of a weak
solution which conserves circulation at the boundary components if the initial
vorticity is integrable. In addition, we discuss the definition of the
mechanical force which the flow exerts on material boundary components and its
relation with conservation of circulation. Finally, we describe the
corresponding results for a bounded domain with holes, and the adaptations
required in the proofs.
