Transport equations for Osgood velocity fields
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909659529478144 |
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| author | Fjordholm, Ulrik Skre Mæhlen, Ola Isaac Høgåsen |
| author_facet | Fjordholm, Ulrik Skre Mæhlen, Ola Isaac Høgåsen |
| contents | We consider the transport equation with a velocity field satisfying the Osgood condition. The weak formulation is not meaningful in the usual Lebesgue sense, meaning that the usual DiPerna--Lions treatment of the problem is not applicable {(in particular, the divergence of the velocity might be unbounded)}. Instead, we use Riemann--Stieltjes integration to interpret the weak formulation, leading to a well-posedness theory in regimes not covered by existing works. The most general results are for the one-dimensional problem, with generalisations to multiple dimensions in the particular case of log-Lipschitz velocities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_11531 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transport equations for Osgood velocity fields Fjordholm, Ulrik Skre Mæhlen, Ola Isaac Høgåsen Analysis of PDEs We consider the transport equation with a velocity field satisfying the Osgood condition. The weak formulation is not meaningful in the usual Lebesgue sense, meaning that the usual DiPerna--Lions treatment of the problem is not applicable {(in particular, the divergence of the velocity might be unbounded)}. Instead, we use Riemann--Stieltjes integration to interpret the weak formulation, leading to a well-posedness theory in regimes not covered by existing works. The most general results are for the one-dimensional problem, with generalisations to multiple dimensions in the particular case of log-Lipschitz velocities. |
| title | Transport equations for Osgood velocity fields |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.11531 |