On solutions of the transport equation in the presence of singularities
Article
Miot, E. and Sharples, N. 2022. On solutions of the transport equation in the presence of singularities. Transactions of the American Mathematics Society. 375 (10), pp. 7187-7207. https://doi.org/10.1090/tran/8701
Type | Article |
---|---|
Title | On solutions of the transport equation in the presence of singularities |
Authors | Miot, E. and Sharples, N. |
Abstract | We consider the transport equation on $[0,T]\times \R^n$ in the situation where the vector field is $BV$ off a set $S\subset [0,T]\times \R^n$. 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. |
Keywords | Transport equation; non-BV singularities; renormalization; avoidance of sets; box-counting dimension; dimension print; vortex dynamics |
Publisher | American Mathematical Society |
Journal | Transactions of the American Mathematics Society |
ISSN | 0002-9947 |
Electronic | 1088-6850 |
Publication dates | |
Online | 29 Jul 2022 |
Oct 2022 | |
Publication process dates | |
Submitted | 24 May 2021 |
Accepted | 01 Mar 2022 |
Deposited | 03 Mar 2022 |
Output status | Published |
Accepted author manuscript | License File Access Level Open |
Copyright Statement | This is the accepted manuscript for an article accepted for publication in Transactions of the American Mathematics Society. This author's accepted manuscript version is made available as permitted by the publisher's (American Mathematical Society) Article Sharing Policy; reproduced under the CC-BY-NC-ND Creative Commons License. |
Digital Object Identifier (DOI) | https://doi.org/10.1090/tran/8701 |
Web of Science identifier | WOS:000834950900001 |
Language | English |
https://repository.mdx.ac.uk/item/89v1y
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