Weak solutions to the parabolic $p$-Laplace equation in a moving domain under a Neumann type boundary condition
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917229239468032 |
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| author | Miura, Tatsu-Hiko |
| author_facet | Miura, Tatsu-Hiko |
| contents | This paper studies the parabolic $p$-Laplace equation with $p>2$ in a moving domain under a Neumann type boundary condition corresponding to the total mass conservation. We establish the existence and uniqueness of a weak solution by the Galerkin method in evolving Bochner spaces and a monotonicity argument. The main difficulty is in characterizing the weak limit of the nonlinear gradient term, where we need to deal with a term which comes from the boundary condition and cannot be absorbed into a monotone operator. To overcome this difficulty, we prove a uniform-in-time Friedrichs type inequality on a moving domain with time-dependent basis functions and make use of it to get the strong convergence of approximate solutions. We also show that the time derivative exists in the $L^2$ sense when given data have a better regularity, and discuss extension of the existence and uniqueness results to a Leray-Lions type operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12598 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak solutions to the parabolic $p$-Laplace equation in a moving domain under a Neumann type boundary condition Miura, Tatsu-Hiko Analysis of PDEs 35K20, 35K92, 35R37 This paper studies the parabolic $p$-Laplace equation with $p>2$ in a moving domain under a Neumann type boundary condition corresponding to the total mass conservation. We establish the existence and uniqueness of a weak solution by the Galerkin method in evolving Bochner spaces and a monotonicity argument. The main difficulty is in characterizing the weak limit of the nonlinear gradient term, where we need to deal with a term which comes from the boundary condition and cannot be absorbed into a monotone operator. To overcome this difficulty, we prove a uniform-in-time Friedrichs type inequality on a moving domain with time-dependent basis functions and make use of it to get the strong convergence of approximate solutions. We also show that the time derivative exists in the $L^2$ sense when given data have a better regularity, and discuss extension of the existence and uniqueness results to a Leray-Lions type operator. |
| title | Weak solutions to the parabolic $p$-Laplace equation in a moving domain under a Neumann type boundary condition |
| topic | Analysis of PDEs 35K20, 35K92, 35R37 |
| url | https://arxiv.org/abs/2505.12598 |