A convex integration scheme for the continuity equation past the Sobolev embedding threshold
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909565435510784 |
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| author | Colombo, Maria Colombo, Roberto Kumar, Anuj |
| author_facet | Colombo, Maria Colombo, Roberto Kumar, Anuj |
| contents | We introduce a convex integration scheme for the continuity equation in the context of the Di Perna-Lions theory that allows to build incompressible vector fields in $C_{t}W^{1,p}_x$ and nonunique solutions in $C_{t} L^{q}_x$ for any $p,q$ with $\frac{1}{p} + \frac{1}{q} > 1 + \frac{1}{d}- δ$ for some $δ>0$. This improves the previous bound, corresponding to $δ=0$, or equivalently $q' > p^*$, obtained with convex integration so far, and critical for those schemes in view of the Sobolev embedding that guarantees that solutions are distributional in the opposite range. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_03578 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A convex integration scheme for the continuity equation past the Sobolev embedding threshold Colombo, Maria Colombo, Roberto Kumar, Anuj Analysis of PDEs 35A02, 35F10 We introduce a convex integration scheme for the continuity equation in the context of the Di Perna-Lions theory that allows to build incompressible vector fields in $C_{t}W^{1,p}_x$ and nonunique solutions in $C_{t} L^{q}_x$ for any $p,q$ with $\frac{1}{p} + \frac{1}{q} > 1 + \frac{1}{d}- δ$ for some $δ>0$. This improves the previous bound, corresponding to $δ=0$, or equivalently $q' > p^*$, obtained with convex integration so far, and critical for those schemes in view of the Sobolev embedding that guarantees that solutions are distributional in the opposite range. |
| title | A convex integration scheme for the continuity equation past the Sobolev embedding threshold |
| topic | Analysis of PDEs 35A02, 35F10 |
| url | https://arxiv.org/abs/2504.03578 |