2023/10/04 by Max Sauerbrey, Sauerbrey, Max
Physics and Astronomy · #35R60 #76A20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2310.02765
openalex publication_date 2023/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ∂x(u^(n)/(2)W) becomes linear. In the case of a non-quadratic mobility exponent, results are only available in the situation that n≥ (8)/(3) leaving the interval of mobility exponents n∈ (2,(8)/(3)) untreated. In this article we resolve the current gap in the literature by presenting a proof, which works under the assumption n∈ (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.