Grün G, Klein L (2025)
Publication Type: Journal article
Publication year: 2025
Book Volume: 25
Article Number: 85
Journal Issue: 3
DOI: 10.1007/s00028-025-01099-1
For mobility exponents n∈(2,3), we prove the existence of strictly positive solutions to stochastic thin-film equations with singular effective interface potential and Stratonovich-type lower-order terms. With the perspective of using these solutions in Part II to construct surface-tension energy-dissipating solutions to stochastic thin-film equations with compactly supported initial data, for which finite speed of propagation is shown in Grün and Klein (SIAM J Math Anal), we establish decay estimates on the sum of surface-tension energy and effective interface potential—without relying on further functionals involving initial data. Besides an extension of earlier techniques used in the case n=2 and a refinement of oscillation estimates for discrete solutions, the main analytical novelty of this paper is a discretization method, which shows nonnegativity for a finite-element counterpart of the integral ∫O(un-2ux)xuxxdx under periodic boundary conditions in the parameter regime n∈(2,3). This nonnegativity property serves to control Itô-correction terms in the estimate for the decay of the surface-tension energy. This way, it is the key to obtain the desired decay estimates for the sum of surface-tension energy and effective interface potential, which permit to establish the singular limit of vanishing effective interface potential in Part II.
APA:
Grün, G., & Klein, L. (2025). Existence of nonnegative energy-dissipating solutions to a class of stochastic thin-film equations under weak slippage: part I—positive solutions. Journal of Evolution Equations, 25(3). https://doi.org/10.1007/s00028-025-01099-1
MLA:
Grün, Günther, and Lorenz Klein. "Existence of nonnegative energy-dissipating solutions to a class of stochastic thin-film equations under weak slippage: part I—positive solutions." Journal of Evolution Equations 25.3 (2025).
BibTeX: Download