Well-posedness of the stochastic thin-film equation with an interface potential

Antonio Agresti; Max Sauerbrey

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arXiv · 2403.12652

Well-posedness of the stochastic thin-film equation with an interface potential

Antonio Agresti·Max Sauerbrey

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Abstract

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 $\alpha$-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.

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Antonio Agresti, Max Sauerbrey. 2024-03-19. Well-posedness of the stochastic thin-film equation with an interface potential. https://arxiv.org/abs/2403.12652

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