Non-branching properties of Regular Lagrangian Flows and well posedness of the continuity equation in the class of nonnegative solutions
Abstract
In this paper we deal with the theory of Regular Lagrangian Flows, as initiated in [29] and axiomatized in [2]. In the first part of the paper we show in Theorem 4.3 (see also Corollary 4.5) that the decomposition of ℒ1+d-almost all the space time in a family of (essentially) disjoint characteristics, the technique at the basis of the well posedness in [1] (for planar, divergence-free and autonomous vector fields), and in [13] (for vector fields having a suitable regularity in space, including BV vector fields) is not only sufficient for well posedness, but also necessary. This shows that in many cases the typical uniqueness statement of the DiPerna-Lions theory, namely that the path 𝕏(·,x) is (essentially) unique given the initial position x, can, as in the classical theory, be strengthened by saying that the paths 𝕏(·,x) (essentially) do not branch. As the simple Example 1.1 shows, when we consider vector fields with singular divergence well posedness results may hold only for solutions that do not change sign. In the second part of the paper we revisit [8,9] obtaining a well posedness result for special vector fields of bounded deformation and nonnegative solutions, under a mild regularity assumption (5.3) with respect to the time variable, weaker than the space-time SBD regularity assumption made in [9] for nearly incompressible vector fields. Finally, we revisit carefully the constructions of Section 9 of [13], to see how they provide existence and uniqueness for (1.2) assuming BV regularity in space, in the class of nonnegative bounded solutions.
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BibTeX
@misc{ambrosio2026nonbranching,
title = {{Non-branching properties of Regular Lagrangian Flows and well posedness of the continuity equation in the class of nonnegative solutions}},
author = {Luigi Ambrosio and Stefano Bianchini and Federico Renzi and Federico Vitillaro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17129},
eprint = {2610.17129},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17129},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-branching properties of Regular Lagrangian Flows and well posedness of the continuity equation in the class of nonnegative solutions
AU - Luigi Ambrosio
AU - Stefano Bianchini
AU - Federico Renzi
AU - Federico Vitillaro
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17129
UR - https://arxiv.org/abs/2610.17129
AB - In this paper we deal with the theory of Regular Lagrangian Flows, as initiated in [29] and axiomatized in [2]. In the first part of the paper we show in Theorem 4.3 (see also Corollary 4.5) that the decomposition of ℒ^1+d-almost all the space time in a family of (essentially) disjoint characteristics, the technique at the basis of the well posedness in [1] (for planar, divergence-free and autonomous vector fields), and in [13] (for vector fields having a suitable regularity in space, including BV vector fields) is not only sufficient for well posedness, but also necessary. This shows that in many cases the typical uniqueness statement of the DiPerna-Lions theory, namely that the path 𝕏(·,x) is (essentially) unique given the initial position x, can, as in the classical theory, be strengthened by saying that the paths 𝕏(·,x) (essentially) do not branch. As the simple Example 1.1 shows, when we consider vector fields with singular divergence well posedness results may hold only for solutions that do not change sign. In the second part of the paper we revisit [8,9] obtaining a well posedness result for special vector fields of bounded deformation and nonnegative solutions, under a mild regularity assumption (5.3) with respect to the time variable, weaker than the space-time SBD regularity assumption made in [9] for nearly incompressible vector fields. Finally, we revisit carefully the constructions of Section 9 of [13], to see how they provide existence and uniqueness for (1.2) assuming BV regularity in space, in the class of nonnegative bounded solutions.
ER -