Long Time Behavior of Stochastic Thin Film Equation

Abstract: We consider the stochastic thin-film equation with linear deterministic and stochastic Itô perturbations. The existence of nonnegative weak martingale solutions on the semi-axis is established, and their asymptotic behavior as $t \to \infty$ is investigated. It is shown that in square mean the $L<sup>\infty$ norm of the solution converges to the spatial mean value of the initial condition, multiplied by a random factor similar to a geometric Wiener process.

• The paper establishes global nonnegative weak martingale solutions for stochastic thin film equations using Trotter-Kato splitting and entropy methods.
• It characterizes long-time dynamics by identifying conditions under which noise induces extinction or yields a spatially uniform quasi-equilibrium profile.
• The analysis quantifies mass evolution via a stochastic exponential, demonstrating that strong noise can override deterministic growth to drive film decay.

Long-Time Asymptotics for the Stochastic Thin Film Equation

Introduction

This work establishes the global existence and long-time asymptotic behavior for a class of stochastic thin film equations (STFEs) incorporating both linear deterministic forcing and multiplicative Itô noise. These fourth-order degenerate parabolic equations model the evolution of the height profile $u(x,t)$ of a thin liquid film spreading on a substrate, with mobility $u^n$, periodic boundary conditions, and nonnegative initial data. The principal focus is on the regime $n \ge 4$, which is of physical and mathematical relevance due to strong degeneracy near $u=0$ and the absence of a maximum principle.

Problem Formulation and Context

The stochastic thin film equation considered is:

$$\mathrm{d}u = -\partial_x(u^n \partial_x^3 u) \, \mathrm{d}t + \gamma(t) u \, \mathrm{d}t + \alpha(t) u \, \mathrm{d}\beta(t)$$

on $x \in [0,L]$, $t \ge 0$, with periodic boundary and nonnegative initial data $u_0$. Here, $\beta(t)$ is a standard Wiener process and $\gamma(t)$, $u^n$0 are continuous coefficients representing deterministic and stochastic perturbations, respectively.

While deterministic thin film equations are well-studied, the stochastic variants exhibit challenging analytical behavior, including mass loss/gain, positivity/nonnegativity concerns, and subtle interactions between noise and degeneracy. Existing literature on SPDEs for thin films has focused primarily on existence of weak or martingale solutions under various regularizations and mobilities [Gess & Gnann (2020), Fischer & Grün (2018)], but has addressed large-time asymptotics only peripherally.

Main Contributions

Existence and Positivity

The paper first rigorously constructs globally defined, nonnegative weak martingale solutions for the STFEs, for all $u^n$1 (Theorem 2.1 and 2.2). The analysis proceeds via a Trotter-Kato operator splitting, alternately evolving deterministic and stochastic flows on small time intervals. Uniform a priori bounds are recovered using entropy methods inspired by Barenblatt-type solutions: specifically, $u^n$2-norm estimates for $u^n$3 provide crucial control in the degenerate regime.

For $u^n$4 and strictly positive initial data, positivity propagates in time almost surely, despite the strong degeneracy (Proposition 5.1). A contradiction argument using local Hölder continuity and divergence of the $u^n$5 moment ensures that $u^n$6 cannot attain zero on any subset of positive measure. This enables extending weak solution concepts to include the full domain, not just the strictly positive set.

Asymptotic Behavior

The large-time dynamics are characterized according to the asymptotics of the mass evolution equation:

$u^n$7

where $u^n$8 is a geometric-type stochastic process.

The main results are:

• Extinction regime (Theorem 2.3): If $u^n$9 and $n \ge 4$0, or under the pathwise Lyapunov-type condition

$n \ge 4$1

then $n \ge 4$2 both in the mean and almost surely. Notably, this dissipation/extinction can occur even for positive $n \ge 4$3, provided the noise is sufficiently strong.

• Quasi-equilibrium regime (Theorem 2.5): If both $n \ge 4$4 and $n \ge 4$5 are $n \ge 4$6-integrable and $n \ge 4$7, then the solution approaches a spatially constant random profile:

$n \ge 4$8

where $n \ge 4$9 is the spatial average of the initial data. The limiting weight $u=0$0 is random, capturing the annealed effect of the stochastic mass fluctuations.

These results explicitly quantify the impact of time-dependent stochastic and deterministic growth/dissipation terms on film persistence or extinction. The proofs rely on Khasminskii's stability theory for linear SDEs, $u=0$1-energy and entropy inequalities, and fine moment calculations for the stochastic exponentials $u=0$2.

Methodology and Analysis

The analysis leverages:

• Trotter-Kato Splitting: The equation is split into deterministic high-order degenerate PDE and linear stochastic (multiplicative) ODE steps, with careful control at the interface.
• Entropy and Energy Estimates: Multiplicative Itô noise preserves nonnegativity while introducing delicate mass fluctuations. Elliptic and parabolic regularization arguments track higher moments and Sobolev norms, enabling passage to the limit and establishing global solutions.
• Khasminskii's Criteria: For the limiting mass evolution, explicit necessary and sufficient conditions for almost sure extinction/survival are invoked, transferring the large-time dynamics to the solution of a corresponding linear multiplicative SDE.
• Propagation of Positivity: For large $u=0$3, entropy bounds preclude development of "dead zones" with zero solution—critical due to the degeneracy at $u=0$4.

The expected exponential decay rates of energy-type functionals are shown to be preserved in the presence of sufficiently weak noise, while strong noise or strong damping enforces extinction regardless of mass injection.

Implications and Perspectives

The results delineate sharp regimes of persistence versus extinction for the stochastic thin film equation, highlighting how mass balance interacts with stochastic dissipation or amplification. One notable consequence is that sufficiently intense noise can override deterministic mass increase and drive extinction, contradicting deterministic intuition. Conversely, weak noise can preserve (randomly scaled) spatial means, but pure self-averaging does not occur—the stochastic weight remains as a random multiplicative factor even asymptotically.

Theoretical implications include establishing a framework for understanding invariant measures (when these exist), large deviation phenomena, and long-time mixing in high-degeneracy SPDEs with multiplicative noise. Practically, these results inform the modeling of thin liquid films in technologically relevant contexts subject to random environmental or active fluctuations.

Future Directions

Potential extensions include:

• Analysis of more general nonlinear stochastic drift/absorption terms, where both mass and higher moments are directly coupled to spatial heterogeneities.
• Higher-dimensional generalizations and coupling with additional physical fields (e.g., surfactant concentration, temperature).
• Investigation of invariant measures and ergodic properties in regimes with persistent noise.
• Incorporation of non-Gaussian or non-Itô noise, where the pathwise analysis may differ substantially.

Conclusion

This paper rigorously demonstrates the existence, positivity, and detailed dichotomy of long-time behavior for the stochastic thin film equation with multiplicative Itô noise. The findings reveal that asymptotic extinction or convergence to a spatially uniform (but randomly weighted) profile are determined by the interplay of deterministic and stochastic growth/dissipation rates, with critical roles for both energy and entropy-based estimates. These results significantly advance the theoretical understanding of degenerate, stochastic, high-order evolution equations modeling thin film flows (2604.10010).