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Well-Posedness of the Stochastic Thin-Film Equation with an Interface Potential
Antonio Agresti1,2, Max Sauerbrey1,3
1Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands.
Communications in Mathematical Physics
|July 11, 2026
Summary
This study analyzes fourth-order conservative quasilinear stochastic partial differential equations, proving local well-posedness and regularization for the stochastic thin-film equation under minimal assumptions.
Area of Science:
- Stochastic Partial Differential Equations (SPDEs)
- Mathematical Physics
- Fluid Dynamics
Background:
- Fourth-order conservative quasilinear SPDEs model phenomena like the stochastic thin-film equation.
- Understanding the behavior of these equations is crucial for various scientific applications.
- Strictly positive solutions are of particular interest.
Purpose of the Study:
- To analyze strictly positive solutions to a class of fourth-order conservative quasilinear SPDEs.
- To establish local well-posedness and regularization properties for the stochastic thin-film equation.
- To investigate global well-posedness in the one-dimensional case.
Main Methods:
- Proving local Lipschitz estimates in Bessel potential spaces.
- Deriving stochastic maximal L^p-regularity estimates for thin-film type operators.
- Utilizing alpha-entropy and energy estimates for the one-dimensional case.
Main Results:
- Established local well-posedness for the stochastic thin-film equation.
- Demonstrated blow-up criteria and instantaneous regularization of solutions.
- Achieved global well-posedness in one dimension under specific conditions.
- Allowed for a broad range of mobility functions, including power laws up to n=6.
Conclusions:
- The study provides a rigorous mathematical framework for analyzing stochastic thin-film equations.
- The results contribute to the understanding of blow-up phenomena and regularization in SPDEs.
- Global well-posedness is established in the one-dimensional case, offering insights into long-term behavior.
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