On existence and uniqueness for transport equations with non-smooth velocity fields under inhomogeneous Dirichlet data
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909463737270272 |
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| author | Eto, Tokuhiro Giga, Yoshikazu |
| author_facet | Eto, Tokuhiro Giga, Yoshikazu |
| contents | A transport equation with a non-smooth velocity field is considered under inhomogeneous Dirichlet boundary conditions. The spatial gradient of the velocity field is assumed in $L^{p'}$ in space and the divergence of the velocity field is assumed to be bounded. By introducing a suitable notion of solutions, it is shown that there exists a unique renormalized weak solution for $L^p$ initial and boundary data for $1/p+1/p'=1$. Our theory is considered as a natural extension of the theory due to DiPerna and Lions (1989), where there is no boundary. Although a smooth domain is considered, it is allowed to be unbounded. A key step is a mollification of a solution. In our theory, mollification in the direction normal to the boundary is tailored to approximate the boundary data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12575 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On existence and uniqueness for transport equations with non-smooth velocity fields under inhomogeneous Dirichlet data Eto, Tokuhiro Giga, Yoshikazu Analysis of PDEs 35Q49 A transport equation with a non-smooth velocity field is considered under inhomogeneous Dirichlet boundary conditions. The spatial gradient of the velocity field is assumed in $L^{p'}$ in space and the divergence of the velocity field is assumed to be bounded. By introducing a suitable notion of solutions, it is shown that there exists a unique renormalized weak solution for $L^p$ initial and boundary data for $1/p+1/p'=1$. Our theory is considered as a natural extension of the theory due to DiPerna and Lions (1989), where there is no boundary. Although a smooth domain is considered, it is allowed to be unbounded. A key step is a mollification of a solution. In our theory, mollification in the direction normal to the boundary is tailored to approximate the boundary data. |
| title | On existence and uniqueness for transport equations with non-smooth velocity fields under inhomogeneous Dirichlet data |
| topic | Analysis of PDEs 35Q49 |
| url | https://arxiv.org/abs/2501.12575 |