Well-posedness of the stochastic thin-film equation with an interface potential
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909984613203968 |
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| author | Agresti, Antonio Sauerbrey, Max |
| author_facet | Agresti, Antonio Sauerbrey, Max |
| contents | We consider strictly positive solutions to a class of fourth-order conservative quasilinear SPDEs on the $d$-dimensional torus modeled after the stochastic thin-film equation. We prove local Lipschitz estimates in Bessel potential spaces under minimal assumptions on the parameters and corresponding stochastic maximal $L^p$-regularity estimates for thin-film type operators with measurable in-time coefficients. As a result, we deduce local well-posedness of the stochastic thin-film equation as well as blow-up criteria and instantaneous regularization for the solution. In dimension one, we additionally close $α$-entropy estimates and subsequently an energy estimate for the stochastic thin-film equation with an interface potential so that global well-posedness follows. We allow for a wide range of mobility functions including the power laws $u^n$ for $n\in [0,6)$ as long as the interface potential is sufficiently repulsive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12652 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Well-posedness of the stochastic thin-film equation with an interface potential Agresti, Antonio Sauerbrey, Max Analysis of PDEs Probability 35R60, 76A20, 35K59 We consider strictly positive solutions to a class of fourth-order conservative quasilinear SPDEs on the $d$-dimensional torus modeled after the stochastic thin-film equation. We prove local Lipschitz estimates in Bessel potential spaces under minimal assumptions on the parameters and corresponding stochastic maximal $L^p$-regularity estimates for thin-film type operators with measurable in-time coefficients. As a result, we deduce local well-posedness of the stochastic thin-film equation as well as blow-up criteria and instantaneous regularization for the solution. In dimension one, we additionally close $α$-entropy estimates and subsequently an energy estimate for the stochastic thin-film equation with an interface potential so that global well-posedness follows. We allow for a wide range of mobility functions including the power laws $u^n$ for $n\in [0,6)$ as long as the interface potential is sufficiently repulsive. |
| title | Well-posedness of the stochastic thin-film equation with an interface potential |
| topic | Analysis of PDEs Probability 35R60, 76A20, 35K59 |
| url | https://arxiv.org/abs/2403.12652 |