Viscous Thin Films Driven by Thermal Noise
This talk explores the mathematical analysis of stochastic thin-film equations, focusing on the existence of solutions for nonlinear noise and measure-valued solutions, highlighting the importance of the Stratonovich formulation for accurate physical modelling of capillary-driven viscous thin films at microscopic scales.
Overview
In this talk, we investigate mathematically how capillary-driven viscous thin fluid films evolve on microscopic length scales, in which case thermal noise due to fluctuations of the fluid particles comes into play. The underlying stochastic partial differential equation (SPDE) is a stochastic thin-film equation, a fourth-order degenerate-paraolic PDE driven by nonlinear gradient noise. This equation was first suggested in the physical literature less than 20 years ago and existence of solutions for nonlinear noise was only established very recently. The key observation is that the Stratonovich formulation of the equation is the physically correct mathematical formulation, leading to a suitable balance of fluctuations and dissipation of the underlying physics and the correct balance in the energy-entropy dissipation relations. Specifically we are able to establish existence of nonnegative martingale solutions for nonlinear mobilities and we further prove existence of measure-valued solutions for initial values with non-full support. The latter forms a first step towards proving finite speed of propagation and for investigating contact-line dynamics on microscopic length scales.
This talk is based on joint works with Konstantinos Dareiotis (Leeds), Benjamin Gess (TU Berlin and Max Planck Institute MiS, Leipzig), Günther Grün (Erlangen), and Max Sauerbrey (formerly TU Delft, now Max Planck Institute MiS, Leipzig).
Presenters
Manuel V. Gnann, Assistant Professor, Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Netherlands
Brief Biography
Manuel Gnann obtained Diplom degrees (equivalent MSc) in physics and mathematics in 2009 and 2010 from the University of Konstanz and a PhD degree in mathematics in 2014 prepared at the Max Planck Institute for Mathematics in the Sciences in Leipzig under the supervision of Felix Otto. He was then Fields Postdoctoral Fellow at the Fields Institute in Toronto under the mentorship of Robert McCann and moved to an assistant professorship at the University of Michigan, Ann Arbor. After an appointment in industry, postdocs at the Max Planck Institute in Leipzig and the Technical University of Munich under the mentorship of Christian Kuehn, and an acting professorship at Heidelberg University, he became tenure-track assistant professor at Delft University of Technology in 2019 and was tenured in 2022.
Manuel Gnann’s research is on the applied analysis of (stochastic) partial differential equations in science and technology, with a focus on equations from fluid mechanics and free-boundary problems with boundary contact (codimension-two free-boundary problems). His research was awarded with a couple of grants and awards, among which most recently a Vidi Grant of the Dutch Research Council (NWO).