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

Antonio Agresti, Max Sauerbrey

TL;DR

This work establishes a rigorous local well-posedness theory for stochastic thin-film-type equations with degenerate fourth-order operators on the torus, using stochastic maximal $L^p$-regularity and a carefully regularized framework. It proves local existence and blow-up criteria in any dimension, and develops instantaneous regularization that enables parameter-independent blow-up control. In one dimension, the authors derive $\boldsymbol{\alpha}$-entropy and energy estimates to obtain global well-posedness for a broad mobility class including $m(u)=u^n$ with $n<6$ under repulsive interface potentials, and treat both Itô and Stratonovich noises. The approach hinges on transferring regularized results to the original equation, preserving positivity, and providing sharp a priori bounds that close global-in-time behavior in 1D. These results advance understanding of stochastic degenerate parabolic PDEs arising in thin-film dynamics and offer a solid foundation for numerical schemes and further analysis of contact-line and interface phenomena.

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 $α$-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.

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

TL;DR

This work establishes a rigorous local well-posedness theory for stochastic thin-film-type equations with degenerate fourth-order operators on the torus, using stochastic maximal -regularity and a carefully regularized framework. It proves local existence and blow-up criteria in any dimension, and develops instantaneous regularization that enables parameter-independent blow-up control. In one dimension, the authors derive -entropy and energy estimates to obtain global well-posedness for a broad mobility class including with under repulsive interface potentials, and treat both Itô and Stratonovich noises. The approach hinges on transferring regularized results to the original equation, preserving positivity, and providing sharp a priori bounds that close global-in-time behavior in 1D. These results advance understanding of stochastic degenerate parabolic PDEs arising in thin-film dynamics and offer a solid foundation for numerical schemes and further analysis of contact-line and interface phenomena.

Abstract

We consider strictly positive solutions to a class of fourth-order conservative quasilinear SPDEs on the -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 -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 -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 for as long as the interface potential is sufficiently repulsive.
Paper Structure (19 sections, 26 theorems, 303 equations)

This paper contains 19 sections, 26 theorems, 303 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. Local well-posedness, regularity and blow-up criteria in any dimension
  3. Global well-posedness in one dimension
  4. Further comments on the literature
  5. Notation
  6. Local well-posedness of thin-film type equations in any dimension
  7. Local Lipschitzianity of the regularized coefficients
  8. Stochastic maximal regularity of thin-film type operators
  9. Local well-posedness and blow-up criteria I -- Regularized problem
  10. Instantaneous regularization and blow-up criteria II -- Regularized problem
  11. Transference to the original equation
  12. Global well-posedness in one dimension
  13. $\alpha$-entropy estimates
  14. Proof of the energy estimate -- Lemma \ref{['Lemma_Energy_Est']}
  15. Some functional analytic tools
  16. ...and 4 more sections

Key Result

Theorem 1.6

Let Assumptions Assumptions_coefficients and Assumptions_noise_local with $(s_{\psi},q_{\psi})=(s,q)$ be satisfied, and Then there exists a positive maximal unique $(p,\kappa,s,q)$-solution $(u,\sigma)$ to Eq101 such that a.s. $\sigma>0$ and for all $\theta\in [0,\frac{1}{2})$, if $p>2$.

Theorems & Definitions (55)

  • Remark 1.2
  • Definition 1.3: Local solution
  • Definition 1.4: Maximal unique positive solution
  • Theorem 1.6: Local well-posedness
  • Proposition 1.7: Instantaneous regularization
  • Proposition 1.8: Blow-up criteria
  • Theorem 1.12: Global well-posedness in one dimension -- Itô
  • Theorem 1.13: Global well-posedness in one dimension -- Stratonovich
  • Lemma 2.1
  • proof
  • ...and 45 more