Normal traces and applications to continuity equations on bounded domains

We study several properties of the normal Lebesgue trace of vector fields introduced by the second and third author, De Rosa and Inversi (2024), in the context of the energy conservation for the Euler equations in Onsager-critical classes.Among other things, we prove that the normal Lebesgue trace satisfies the Gauss-Green identity and, by providing explicit counterexamples, that it is a notion sitting strictly between the distributional one for measure-divergence vector fields and the strong one for BV functions.These results are then applied to the study of the uniqueness of weak solutions for continuity equations on bounded domains, allowing for the removal of the assumption in Crippa et el.(2014a) of global BV regularity up to the boundary, at least around the portion of the boundary where the characteristics exit the domain or are tangent.The proof relies on an explicit renormalization formula completely characterized by the boundary datum and the positive part of the normal Lebesgue trace.In the case when the characteristics enter the domain, a counterexample shows that achieving the normal trace in the Lebesgue sense is not enough to prevent nonuniqueness, and thus a BV assumption seems to be necessary to get uniqueness. 1 In the literature the notation DM p ( ) is probably more common, i.e., "divergence-measure" vector fields.