On well-posedness for parabolic Cauchy problems of Lions type with rough initial data
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914366456070144 |
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| author | Auscher, Pascal Hou, Hedong |
| author_facet | Auscher, Pascal Hou, Hedong |
| contents | We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a range of $p$ for which tempered distributions in homogeneous Hardy--Sobolev spaces $\dot{H}^{s,p}$ with regularity index $s \in (-1,1)$ are initial data. Source terms of Lions' type lie in weighted tent spaces, and weak solutions are built with their gradients in weighted tent spaces as well. A similar result can be achieved for initial data in homogeneous Besov spaces $\dot{B}^{s}_{p,p}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15775 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On well-posedness for parabolic Cauchy problems of Lions type with rough initial data Auscher, Pascal Hou, Hedong Analysis of PDEs Classical Analysis and ODEs Functional Analysis Primary 35K45, Secondary 42B37, 35K05, 42B30, 46E35 We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a range of $p$ for which tempered distributions in homogeneous Hardy--Sobolev spaces $\dot{H}^{s,p}$ with regularity index $s \in (-1,1)$ are initial data. Source terms of Lions' type lie in weighted tent spaces, and weak solutions are built with their gradients in weighted tent spaces as well. A similar result can be achieved for initial data in homogeneous Besov spaces $\dot{B}^{s}_{p,p}$. |
| title | On well-posedness for parabolic Cauchy problems of Lions type with rough initial data |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis Primary 35K45, Secondary 42B37, 35K05, 42B30, 46E35 |
| url | https://arxiv.org/abs/2406.15775 |