Normal traces and applications to continuity equations on bounded domains
Abstract
In this work, we study several properties of the normal Lebesgue trace of vector fields introduced by the second and third author in [18] in the context of the energy conservation for the Euler equations in Onsagercritical classes. Among several properties, we prove that the normal Lebesgue trace satisfies the GaussGreen identity and, by providing explicit counterexamples, that it is a notion sitting strictly between the distributional one for measuredivergence 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 to remove the assumption in [15] 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 for the uniqueness of weak solutions.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2405.11486
 arXiv:
 arXiv:2405.11486
 Bibcode:
 2024arXiv240511486C
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 27 pages, 2 figures. Comments are welcome!