  We construct infinitely many admissible weak solutions to the 2D
incompressible Euler equations for vortex sheet initial data. Our initial datum
has vorticity concentrated on a simple closed curve in a suitable H\"older
space and the vorticity may not have a distinguished sign. Our solutions are
obtained by means of convex integration; they are smooth outside a "turbulence"
zone which grows linearly in time around the vortex sheet. As a byproduct, this
approach shows how the growth of the turbulence zone is controlled by the local
energy inequality and measures the maximal initial dissipation rate in terms of
the vortex sheet strength.
