## On solutions of the transport equation in the presence of singularities

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- by Evelyne Miot and Nicholas Sharples PDF
- Trans. Amer. Math. Soc.
**375**(2022), 7187-7207

## Abstract:

We consider the transport equation on $[0,T]\times \mathbb {R}^n$ in the situation where the vector field is $BV$ off a set $S\subset [0,T]\times \mathbb {R}^n$. We demonstrate that solutions exist and are unique provided that the set of singularities has a sufficiently small anisotropic fractal dimension and the normal component of the vector field is sufficiently integrable near the singularities. This result improves upon recent results of Ambrosio who requires the vector field to be of bounded variation everywhere.

In addition, we demonstrate that under these conditions almost every trajectory of the associated regular Lagrangian flow does not intersect the set $S$ of singularities.

Finally, we consider the particular case of an initial set of singularities that evolve in time so the singularities consists of curves in the phase space, which is typical in applications such as vortex dynamics. We demonstrate that solutions of the transport equation exist and are unique provided that the box-counting dimension of the singularities is bounded in terms of the Hölder exponent of the curves.

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## Additional Information

**Evelyne Miot**- Affiliation: CNRS and Institut Fourier, Université Grenoble Alpes, France
- MR Author ID: 878324
- Email: evelyne.miot@univ-grenoble-alpes.fr
**Nicholas Sharples**- Affiliation: Middlesex University, United Kingdom
- MR Author ID: 1013695
- ORCID: 0000-0003-1722-5647
- Email: n.sharples@mdx.ac.uk
- Received by editor(s): May 24, 2021
- Received by editor(s) in revised form: January 21, 2022, and February 22, 2022
- Published electronically: July 29, 2022
- Additional Notes: The first author was partially supported by the french Agence Nationale de la Recherche through the following projects: SINGFLOWS (grant ANR-18-CE40-0027-01), and INFAMIE (grant ANR-15-CE40-01)
- © Copyright 2022 by the authors
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 7187-7207 - MSC (2020): Primary 35A01, 35A02, 28A35
- DOI: https://doi.org/10.1090/tran/8701