On well-posedness and maximal regularity for parabolic Cauchy problems on weighted tent spaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910039654006784 |
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| author | Auscher, Pascal Hou, Hedong |
| author_facet | Auscher, Pascal Hou, Hedong |
| contents | We prove well-posedness in weighted tent spaces of weak solutions to the Cauchy problem $\partial_t u - \mathrm{div} A \nabla u = f, u(0)=0$, where the source $f$ also lies in (different) weighted tent spaces, provided the complex coefficient matrix $A$ is bounded, measurable, time-independent, and uniformly elliptic. To achieve this, we extend the theory of singular integral operators on tent spaces via off-diagonal estimates introduced by [arXiv:1112.4292] to obtain estimates on solutions $u$, and also $\nabla u$, $\partial_t u$, and $\mathrm{div} A \nabla u$ in weighted tent spaces, showing at the same time maximal regularity. Uniqueness follows from a different strategy using interior representation for weak solutions and boundary behavior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_04844 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On well-posedness and maximal regularity for parabolic Cauchy problems on weighted tent spaces Auscher, Pascal Hou, Hedong Analysis of PDEs Classical Analysis and ODEs Primary 35K45, Secondary 42B37 We prove well-posedness in weighted tent spaces of weak solutions to the Cauchy problem $\partial_t u - \mathrm{div} A \nabla u = f, u(0)=0$, where the source $f$ also lies in (different) weighted tent spaces, provided the complex coefficient matrix $A$ is bounded, measurable, time-independent, and uniformly elliptic. To achieve this, we extend the theory of singular integral operators on tent spaces via off-diagonal estimates introduced by [arXiv:1112.4292] to obtain estimates on solutions $u$, and also $\nabla u$, $\partial_t u$, and $\mathrm{div} A \nabla u$ in weighted tent spaces, showing at the same time maximal regularity. Uniqueness follows from a different strategy using interior representation for weak solutions and boundary behavior. |
| title | On well-posedness and maximal regularity for parabolic Cauchy problems on weighted tent spaces |
| topic | Analysis of PDEs Classical Analysis and ODEs Primary 35K45, Secondary 42B37 |
| url | https://arxiv.org/abs/2311.04844 |