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arXiv · 2406.08427

Existence of nonnegative energy-dissipating solutions to a class of stochastic thin-film equations under weak slippage: Part II -- compactly supported initial data

Günther Grün·Lorenz Klein
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Abstract

We prove existence of martingale solutions to a class of stochastic thin-film equations for mobility exponents $n \in (2,3)$ and compactly supported initial data. With the perspective to study free-boundary problems related to stochastic thin-film equations in future work, we start from the surface-tension driven stochastic thin-film equation with Stratonovich noise and exploit the regime of coefficients in front of the Stratonovich correction term (which is of porous-media-type) for which energy dissipation can be established. By Bernis inequalities, third order spatial derivatives of appropriate powers of the solution are controlled. Analytically, we rely on approximation with $\mathbb{P}$-almost surely strictly positive solutions and compactness methods based on energy-entropy estimates as well as martingale identification techniques.

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Günther Grün, Lorenz Klein. 2024-06-12. Existence of nonnegative energy-dissipating solutions to a class of stochastic thin-film equations under weak slippage: Part II -- compactly supported initial data. https://arxiv.org/abs/2406.08427

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