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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.13645 |
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| _version_ | 1866929565454041088 |
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| author | Cortopassi, Tommaso |
| author_facet | Cortopassi, Tommaso |
| contents | We consider the gradient flow of the Ambrosio-Tortorelli functional at fixed $ε>0$, proving existence, uniqueness and $L^2 _t (H_x ^2) \cap L^\infty _t (H^1 _x) \cap H^1 _t (L^2 _x) $ regularity in dimension 2. In particular we improve a previous result where such regularity was known only up to a finite number of space time points, which diverged as $ε\to 0$. By employing a different technique for the crucial $L^2 _t (H^2 _x)$ estimates we can see how in fact the desired regularity holds everywhere. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_13645 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Existence, uniqueness and $L^2_t (H_x ^2) \cap L^\infty_t (H^1_x) \cap H^1_t (L^2_x) $ regularity of the gradient flow of the Ambrosio-Tortorelli functional Cortopassi, Tommaso Analysis of PDEs We consider the gradient flow of the Ambrosio-Tortorelli functional at fixed $ε>0$, proving existence, uniqueness and $L^2 _t (H_x ^2) \cap L^\infty _t (H^1 _x) \cap H^1 _t (L^2 _x) $ regularity in dimension 2. In particular we improve a previous result where such regularity was known only up to a finite number of space time points, which diverged as $ε\to 0$. By employing a different technique for the crucial $L^2 _t (H^2 _x)$ estimates we can see how in fact the desired regularity holds everywhere. |
| title | Existence, uniqueness and $L^2_t (H_x ^2) \cap L^\infty_t (H^1_x) \cap H^1_t (L^2_x) $ regularity of the gradient flow of the Ambrosio-Tortorelli functional |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2301.13645 |