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Solutions to the stochastic thin-film equation for the range of mobility exponents $n\in (2,3)$

Max Sauerbrey

TL;DR

This work establishes the existence of martingale solutions for the one-dimensional stochastic thin-film equation with mobility exponent $n\in(2,3)$ under Stratonovich noise on the torus. The authors introduce inhomogeneous mobility regularizations $F_\delta$ (and $F_{\delta,\varepsilon}$) that dominate near $u=0$ to enforce nonnegativity, and crucially exploit a log-entropy (i.e., $\alpha=-1$ entropy) that matches the noise energy to close the stochastic a-priori estimates. A Galerkin scheme is used to construct solutions to regularized problems, followed by a sequence of limit passages ($R\to\infty$, $\varepsilon\to0$, then $\delta\to0$) via stochastic compactness to obtain a weak martingale solution of the original SPDE, which remains strictly positive almost everywhere and satisfies a combined energy–log-entropy bound. This approach extends prior results to the full mobility range $(2,3)$, bridging a gap in the literature and enabling existence results for non-fully-supported initial data through very weak solutions, while providing robust control through entropy and energy estimates.

Abstract

Recently, many existence results for the stochastic thin-film equation were established in the case of a quadratic mobility exponent $n=2$, in which the noise term $\partial_x(u^\frac{n}{2}\mathcal{W})$ becomes linear. In the case of a non-quadratic mobility exponent, results are only available in the situation that $n\ge \frac{8}{3}$ leaving the interval of mobility exponents $n\in (2,\frac{8}{3})$ untreated. In this article we resolve the current gap in the literature by presenting a proof, which works under the assumption $n\in (2,3)$, i.e., the regime of weak slippage. The key idea is to use that the $\log$-entropy dissipation coincides with the energy production due to the noise. To realize this idea, we approximate the stochastic thin-film equation by stochastic thin-film equations with inhomogeneous mobility functions, which behave like a higher power near $0$. As a consequence the approximate solutions are non-negative, which is vital to use the $\log$-entropy estimate.

Solutions to the stochastic thin-film equation for the range of mobility exponents $n\in (2,3)$

TL;DR

This work establishes the existence of martingale solutions for the one-dimensional stochastic thin-film equation with mobility exponent under Stratonovich noise on the torus. The authors introduce inhomogeneous mobility regularizations (and ) that dominate near to enforce nonnegativity, and crucially exploit a log-entropy (i.e., entropy) that matches the noise energy to close the stochastic a-priori estimates. A Galerkin scheme is used to construct solutions to regularized problems, followed by a sequence of limit passages (, , then ) via stochastic compactness to obtain a weak martingale solution of the original SPDE, which remains strictly positive almost everywhere and satisfies a combined energy–log-entropy bound. This approach extends prior results to the full mobility range , bridging a gap in the literature and enabling existence results for non-fully-supported initial data through very weak solutions, while providing robust control through entropy and energy estimates.

Abstract

Recently, many existence results for the stochastic thin-film equation were established in the case of a quadratic mobility exponent , in which the noise term becomes linear. In the case of a non-quadratic mobility exponent, results are only available in the situation that leaving the interval of mobility exponents untreated. In this article we resolve the current gap in the literature by presenting a proof, which works under the assumption , i.e., the regime of weak slippage. The key idea is to use that the -entropy dissipation coincides with the energy production due to the noise. To realize this idea, we approximate the stochastic thin-film equation by stochastic thin-film equations with inhomogeneous mobility functions, which behave like a higher power near . As a consequence the approximate solutions are non-negative, which is vital to use the -entropy estimate.
Paper Structure (16 sections, 20 theorems, 231 equations)

This paper contains 16 sections, 20 theorems, 231 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Relevant a-priori estimates
  3. Result
  4. Extension of a previous existence result
  5. Strategy of the proof
  6. Notation
  7. Organization of the manuscript
  8. Approximate mobilities and functionals
  9. Construction of solutions
  10. Galerkin approximation
  11. The limits R to infinity and epsilon to zero
  12. delta-uniform estimates
  13. The limit delta to zero
  14. Proofs omitted in the previous section
  15. Technical estimates on the approximate mobilities and functionals
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $T\in (0,\infty)$, $n\in (2,3)$, $p>n+2$ and $u_0\in L^p(\Omega, \mathfrak{F}_0{\color{black};} H^1(\mathop{\mathrm{\mathbb{T}}}\nolimits))$ non-negative with and moreover Then Eq102 admits a weak martingale solution in the sense of Definition defi_sol with initial value $u_0$ satisfying $\tilde{u}>0$,$\tilde{\mathop{\mathrm{\mathbb{P}}}\nolimits}\otimes dt\otimes dx$-almost everywhere. Mor

Theorems & Definitions (44)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Definition 1.4
  • Corollary 1.5
  • Definition 4.1
  • Lemma 4.2: Existence for \ref{['Eq108']}
  • proof
  • Lemma 4.3: $(R,\epsilon,\delta)$-Uniform Entropy Estimate
  • proof
  • ...and 34 more