A remark on selection of solutions for the transport equation
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912027635613696 |
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| author | Pitcho, Jules |
| author_facet | Pitcho, Jules |
| contents | We prove that for bounded, divergence-free vector fields in $L^1_{loc}((0,+\infty);BV_{loc}(R^d;R^d))$, regularisation by convolution of the vector field selects a single solution of the transport equation for any integrable initial datum. We recall the vector field constructed by Depauw in [10], which lies in the above class of vector fields. We show that the transport equation along this vector field has at least two bounded weak solutions for any bounded initial datum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_17085 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A remark on selection of solutions for the transport equation Pitcho, Jules Analysis of PDEs 35A02 - 35D30 - 35Q49 -34A12 We prove that for bounded, divergence-free vector fields in $L^1_{loc}((0,+\infty);BV_{loc}(R^d;R^d))$, regularisation by convolution of the vector field selects a single solution of the transport equation for any integrable initial datum. We recall the vector field constructed by Depauw in [10], which lies in the above class of vector fields. We show that the transport equation along this vector field has at least two bounded weak solutions for any bounded initial datum. |
| title | A remark on selection of solutions for the transport equation |
| topic | Analysis of PDEs 35A02 - 35D30 - 35Q49 -34A12 |
| url | https://arxiv.org/abs/2312.17085 |