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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/2306.06868 |
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| _version_ | 1866912198442352640 |
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| author | Tsubouchi, Shuntaro |
| author_facet | Tsubouchi, Shuntaro |
| contents | We consider weak solutions to very singular parabolic equations involving a one-Laplace-type operator, which is singular and degenerate, and a $p$-Laplace-type operator with $\frac{2n}{n+2}<p<\infty$, where $n\ge 2$ denotes the space dimension. This type of equation is used to describe the motion of a Bingham flow. It has been a long-standing open problem of whether the spatial gradients of weak solutions are continuous in space and time. This paper aims to give an affirmative answer for a wide class of such equations. This equation becomes no longer uniformly parabolic near the facet, the place where the spatial gradient vanishes. To achieve our goal, we show local a priori Hölder continuity of gradients suitably truncated near facets. For this purpose, we consider a parabolic approximate problem and appeal to standard methods, including De Giorgi's truncation and comparisons with Dirichlet heat flows. Our method is a parabolic adjustment of our method developed to prove the corresponding statements for stationary problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_06868 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Continuity of the spatial gradient of weak solutions to very singular parabolic equations involving the one-Laplacian Tsubouchi, Shuntaro Analysis of PDEs 35K92, 35B65, 35A35 We consider weak solutions to very singular parabolic equations involving a one-Laplace-type operator, which is singular and degenerate, and a $p$-Laplace-type operator with $\frac{2n}{n+2}<p<\infty$, where $n\ge 2$ denotes the space dimension. This type of equation is used to describe the motion of a Bingham flow. It has been a long-standing open problem of whether the spatial gradients of weak solutions are continuous in space and time. This paper aims to give an affirmative answer for a wide class of such equations. This equation becomes no longer uniformly parabolic near the facet, the place where the spatial gradient vanishes. To achieve our goal, we show local a priori Hölder continuity of gradients suitably truncated near facets. For this purpose, we consider a parabolic approximate problem and appeal to standard methods, including De Giorgi's truncation and comparisons with Dirichlet heat flows. Our method is a parabolic adjustment of our method developed to prove the corresponding statements for stationary problems. |
| title | Continuity of the spatial gradient of weak solutions to very singular parabolic equations involving the one-Laplacian |
| topic | Analysis of PDEs 35K92, 35B65, 35A35 |
| url | https://arxiv.org/abs/2306.06868 |