Propagation of singularities by Osgood vector fields and for 2D inviscid incompressible fluids

Abstract We show that certain singular structures (Hölderian cusps and mild divergences) are transported by the flow of homeomorphisms generated an Osgood velocity field. The structure these singularities is related to modulus continuity results shown be sharp in sense slightly more cannot generally propagated. For 2D Euler equation, we prove preserved motion, e.g. a system $$\log \log _+(1/|x|)$$ log + ( 1 / | x ) vortices (and those less singular) travel with fluid nonlinear fashion, up bounded perturbations. also give stability for weak solutions away from their set.