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We construct infinitely many admissible weak solutions to the 2Dincompressible Euler equations for vortex sheet initial data. Our initial datumhas vorticity concentrated on a simple closed curve in a suitable H\"olderspace and the vorticity may not have a distinguished sign. Our solutions areobtained by means of convex integration; they are smooth outside a "turbulence"zone which grows linearly in time around the vortex sheet. As a byproduct, thisapproach shows how the growth of the turbulence zone is controlled by the localenergy inequality and measures the maximal initial dissipation rate in terms ofthe vortex sheet strength.

Dissipative Euler flows for vortex sheet initial data without distinguished sign