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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.15529 |
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| _version_ | 1866916122862813184 |
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| author | Bonicatto, Paolo Ciampa, Gennaro Crippa, Gianluca |
| author_facet | Bonicatto, Paolo Ciampa, Gennaro Crippa, Gianluca |
| contents | We study the Cauchy problem for the advection-diffusion equation $\partial_t u + \mathrm{div} (u b ) = Δu$ associated with a merely integrable divergence-free vector field $b$ defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs. We also propose some open problems, motivated by very recent results which show ill-posedness of the equation in certain regimes of integrability via convex integration schemes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15529 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weak and parabolic solutions of advection-diffusion equations with rough velocity field Bonicatto, Paolo Ciampa, Gennaro Crippa, Gianluca Analysis of PDEs We study the Cauchy problem for the advection-diffusion equation $\partial_t u + \mathrm{div} (u b ) = Δu$ associated with a merely integrable divergence-free vector field $b$ defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs. We also propose some open problems, motivated by very recent results which show ill-posedness of the equation in certain regimes of integrability via convex integration schemes. |
| title | Weak and parabolic solutions of advection-diffusion equations with rough velocity field |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2306.15529 |