On the lack of selection for the transport equation over a dense set of vector fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915372617170944 |
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| author | Pitcho, Jules |
| author_facet | Pitcho, Jules |
| contents | We construct a set of bounded vector fields dense in $L^p((0,2);W^{s,p}_{loc}(\mathbb{R}^2;\mathbb{R}^2))$ for $1\leq p <+\infty$ and $0\leq s<1$ with $p<1/s$ for which smooth regularisation of the vector field does not give a selection criterion for the continuity equation, thereby showing that the two examples constructed in [Calc. Var. Partial Differ. Equ. 59 (2019), Ciampa, Crippa and Spirito] and [Ann. Math. Qué. 46 (2022), De Lellis and Giri] are generic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_19687 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the lack of selection for the transport equation over a dense set of vector fields Pitcho, Jules Analysis of PDEs 35A02, 35D30, 35Q49, 34A12 We construct a set of bounded vector fields dense in $L^p((0,2);W^{s,p}_{loc}(\mathbb{R}^2;\mathbb{R}^2))$ for $1\leq p <+\infty$ and $0\leq s<1$ with $p<1/s$ for which smooth regularisation of the vector field does not give a selection criterion for the continuity equation, thereby showing that the two examples constructed in [Calc. Var. Partial Differ. Equ. 59 (2019), Ciampa, Crippa and Spirito] and [Ann. Math. Qué. 46 (2022), De Lellis and Giri] are generic. |
| title | On the lack of selection for the transport equation over a dense set of vector fields |
| topic | Analysis of PDEs 35A02, 35D30, 35Q49, 34A12 |
| url | https://arxiv.org/abs/2404.19687 |