Well-Posedness of the Stochastic Thin-Film Equation with an Interface Potential

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 $$L^p$$ 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 $$\alpha $$ 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 $$u^n$$ u n for $$n\in [0,6)$$ n ∈ [ 0 , 6 ) as long as the interface potential is sufficiently repulsive.