Sobolev regular flows of non-Lipschitz vector fields

Abstract

We show that vector fields with exponentially integrable derivatives admit a well defined flow of homeomorphisms X(t,⋅)∈W1,p(t)loc for some p(t)>1, at least for small times. When the field is certain Riesz potential of a bounded function, the result becomes global in time, due to techniques from Geometric Function Theory. The local result also applies to the flows arising from Yudovich solutions to the planar Euler system with bounded vorticity.