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arXiv · 2610.17129

Non-branching properties of Regular Lagrangian Flows and well posedness of the continuity equation in the class of nonnegative solutions

Luigi Ambrosio·Stefano Bianchini·Federico Renzi·Federico Vitillaro
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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 $\mathcal{L}^{1+d}$-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 $\mathbb{X}(\cdot,x)$ is (essentially) unique given the initial position $x$, can, as in the classical theory, be strengthened by saying that the paths $\mathbb{X}(\cdot,x)$ (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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Luigi Ambrosio, Stefano Bianchini, Federico Renzi, Federico Vitillaro. 2026-10-12. Non-branching properties of Regular Lagrangian Flows and well posedness of the continuity equation in the class of nonnegative solutions. https://arxiv.org/abs/2610.17129

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