2D vorticity Euler equations: Superposition solutions and nonlinear Markov processes

In this note we contribute two results to the theory of the 2D Euler equations in vorticity form on the full plane. First, we establish a generalized Lagrangian representation of weak (in general measure-valued) solutions, which includes and extends classical results on the Lagrangianity of weak solutions. Second, we construct nonlinear Markov processes which are uniquely determined by a selection of weak solutions from initial data in L1∩Lp, p≥2, and related spaces such as the classical and uniformly localized Yudovich space. It is well-known that for p<∞ weak solutions are in general not unique, which renders a suitable selection nontrivial.