Abstract
In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of anL¹function, extending the Lagrangian theory in BouchutCrippa13. The proof is based on a combination of a stability estimate via optimal transport techniques developed in Seis16a and some tools from harmonic analysis introduced in BouchutCrippa13. In the second part of the paper, we address a question that arose in FilhoMazzucatoNussenzveig06, namely whether 2D Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data has low integrability. We show that this is the case even when the initial vorticity is only in L¹ , extending the proof for theLᵖcase in CrippaSpirito15.