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Normal traces and
applications to continuity equations on bounded domains
Gianluca Crippa, Luigi De Rosa, Marco Inversi and Matteo Nesi |
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Vol. 19 (2026), No. 6, 1191–1224
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Abstract |
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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 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 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 assumption seems to be necessary to get uniqueness. |
Keywords
normal traces, continuity equations, uniqueness vs.
nonuniqueness, $BV$ vector fields
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Mathematical Subject Classification
Primary: 28A25, 34A12, 35D30, 35Q49
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Milestones
Received: 17 June 2024
Revised: 3 June 2025
Accepted: 27 August 2025
Published: 13 July 2026
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Authors |
| Gianluca Crippa | |
| Departement Mathematik und
Informatik Universität Basel Basel Switzerland |
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| Luigi De Rosa | |
| Departement Mathematik und
Informatik Universität Basel Basel Switzerland |
Gran Sasso Science Institute L’Aquila Italy |
| Marco Inversi | |
| Department Mathematik und
Informatik Universität Basel Basel Switzerland |
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| Matteo Nesi | |
| Department Mathematik und
Informatik Universität Basel Basel Switzerland |
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| © 2026 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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