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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2409.18900 |
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| _version_ | 1866912588178128896 |
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| author | Córdoba, Diego Martínez-Zoroa, Luis Ożański, Wojciech S. |
| author_facet | Córdoba, Diego Martínez-Zoroa, Luis Ożański, Wojciech S. |
| contents | Given $s\in (3/2,2)$ and $\varepsilon >0$, we construct a compactly supported initial data $θ_0$ such that $\| θ_0 \|_{H^s}\leq \varepsilon$ and there exist $T>0$, $c>0$ and a local-in-time solution $θ$ of the SQG equation that is compactly supported in space, continuous and differentiable in $t$ and in $x$ on $\mathbb{R}^2\times [0,T]$, and, for each $t\in [0,T]$, $ θ(\cdot ,t ) \in {H^{s/(1+ct)}}$ and $ θ(\cdot ,t ) \not \in {H^β}$ for any $β> s/(1+ct)$. Moreover, $θ$ is unique among all solutions with initial condition $θ_0$ which belong to $C([0,T];H^{1+α})$ for any $α>0$ and is continuous and differentiable in $t$ and in $x$ on $\mathbb{R}^2\times [0,T]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18900 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Instantaneous continuous loss of regularity for the SQG equation Córdoba, Diego Martínez-Zoroa, Luis Ożański, Wojciech S. Analysis of PDEs Given $s\in (3/2,2)$ and $\varepsilon >0$, we construct a compactly supported initial data $θ_0$ such that $\| θ_0 \|_{H^s}\leq \varepsilon$ and there exist $T>0$, $c>0$ and a local-in-time solution $θ$ of the SQG equation that is compactly supported in space, continuous and differentiable in $t$ and in $x$ on $\mathbb{R}^2\times [0,T]$, and, for each $t\in [0,T]$, $ θ(\cdot ,t ) \in {H^{s/(1+ct)}}$ and $ θ(\cdot ,t ) \not \in {H^β}$ for any $β> s/(1+ct)$. Moreover, $θ$ is unique among all solutions with initial condition $θ_0$ which belong to $C([0,T];H^{1+α})$ for any $α>0$ and is continuous and differentiable in $t$ and in $x$ on $\mathbb{R}^2\times [0,T]$. |
| title | Instantaneous continuous loss of regularity for the SQG equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.18900 |