2018/11/19 by Cornalba, Federico
#35G20 #35R60 #60H15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1811.07826
Stochastic conservation laws are often challenging when it comes to proving existence of non-negative solutions. In a recent work by J. Fischer and G. Grün (2018, Existence of positive solutions to stochastic thin-film equations, SIAM J. Math. Anal.), existence of positive martingale solutions to a conservative stochastic thin-film equation is established in the case of quadratic mobility. In this work, we focus on a larger class of mobilities (including the linear one) for the thin-film model. In order to do so, we need to introduce nonlinear source potentials, thus obtaining a non-conservative version of the thin-film equation. For this model, we assume the existence of a sufficiently regular local solution (i.e., defined up to a stopping time τ) and, by providing suitable conditions on the source potentials and the noise, we prove that such solution can be extended up to any T>0 and that it is positive with probability one. A thorough comparison with the aforementioned reference work is provided.