Abstract : Thermal fluctuations may influence the evolution of interfaces in multi-phase flow. For this reason, analysis and numerics of stochastic versions of Allen-Cahn or Cahn-Hilliard equations with multiplicative noise, sometimes coupled to momentum equations of fluid dynamics, gained the interest of researchers both in pde and in stochastics. Related to these questions is the investigation of stochastic thin-film equations with their fascinating interplay between degenerate parabolicity and multiplicative noise.
This mini-symposium is supposed to bring researchers together who recently made important contributions both to analysis and numerics of such problems and to foster new collaborations.
[03080] The stochastic Navier-Stokes-Allen-Cahn system with singular potential
Format : Talk at Waseda University
Author(s) :
Andrea Di Primio (Politecnico di Milano)
Maurizio Grasselli (Politecnico di Milano)
Luca Scarpa (Politecnico di Milano)
Abstract : In this talk, I consider the stochastic Navier-Stokes-Allen-Cahn system in a bounded domain of $\mathbb{R}^d$, with $d \in \{2,3\}$, and give some insights on the existence of martingale (in two and three dimensions) and probabilistically-strong solutions (in two dimensions). With respect to its deterministic counterpart, two independent cylindrical stochastic perturbations, which account for thermodynamical effects (e.g., microscopic collisions), are introduced. Moreover, a singular potential is considered, as prescribed by the thermodynamical derivation of the model.
[03081] On some stochastic phase-field models of Cahn-Hilliard-Cook type with logarithmic potential
Format : Online Talk on Zoom
Author(s) :
Luca Scarpa (Politecnico di Milano)
Abstract : We give an overview of some recent results on stochastic phase-field models with logarithmic potential, which cover the celebrated Cahn-Hilliard-Cook equation. Both the conservative and the non-conservative cases are considered, as well as degenerate and non-degenerate mobilities. Well-posedness, regularity, and long-time behaviour of solutions are discussed, with a mention on uniqueness-by-noise. The works presented in the talk are based on joint collaborations with A. Di Primio, Prof. M. Grasselli, and Dr. M. Zanella.
[03044] Temperature Effects in Generalized Diffusions
Format : Talk at Waseda University
Author(s) :
Chun Liu (Illinois Tech)
Abstract : Abstract: In this work, we will introduce a general framework
to derive thermodynamics of a mechanical system, which guarantee
the consistence between the energetic variational approaches with
the laws of thermodynamics. In particular, we will focus on the
coupling between the thermal and mechanical forces. We will also present
some analysis results and difficulties to these systems.
[04783] Asymptotics of the stochastic Cahn-Hilliard equation with space-time white noise
Format : Online Talk on Zoom
Author(s) :
Lubomir Banas (Bielefeld University)
Abstract : We study the sharp interface limit of the stochastic Cahn-Hilliard equation with space-time white noise. We show that for sufficiently strong scaling of the noise the solution of the equation converges to the solution of the deterministic Hele-Shaw problem. We also discuss corresponding results for the numerical approximation of the problem.
[03980] Weak error analysis for the stochastic Allen-Cahn equation
Format : Online Talk on Zoom
Author(s) :
Dominic Breit (TU Clausthal)
Andreas Prohl Tuebingen (University of Tuebingen)
Abstract : We prove strong rate {\em resp.}~weak rate ${\mathcal O}(\tau)$ for a structure preserving temporal discretization (with $\tau$ the step size) of the stochastic Allen-Cahn equation with additive {\em resp.}~multiplicative colored noise in $d=1,2,3$ dimensions.
Direct variational arguments exploit the one-sided Lipschitz property of the cubic nonlinearity in the first setting to settle first order strong rate. It is the same property which allows for uniform bounds for the derivatives of the solution of the related Kolmogorov equation, and then leads to weak rate ${\mathcal O}(\tau)$ in the presence of multiplicative noise. Hence, we obtain twice the rate of convergence known for the strong error in the presence of multiplicative noise.
[03932] On a convergent SAV scheme for stochastic phase-field equations
Format : Talk at Waseda University
Author(s) :
Stefan Metzger (FAU Erlangen-Nürnberg)
Abstract : In this talk, we discuss the numerical treatment of stochastic Cahn-Hilliard equations with stochastic dynamic boundary conditions. These equations can be used to describe contact line tension effects in two-phase flows. By applying a stochastic version of the SAV method, we derive a stable, fully discrete finite element scheme that is linear with respect to the unknown quantities. Furthermore, we establish convergence of the discrete solutions towards martingale solutions using Skorokhod-type arguments.
[03355] Martingale solutions to the stochastic thin-film equation in two dimensions
Format : Talk at Waseda University
Author(s) :
Max Sauerbrey (TU Delft)
Abstract : We construct solutions to the stochastic thin-film equation with quadratic mobility and Stratonovich gradient noise in the physically relevant dimension $d=2$ and allow in particular for solutions with non-full support. The construction relies on a Trotter-Kato time-splitting scheme, which was recently employed in $d=1$. The additional analytical challenges due to the higher spatial dimension are overcome using $\alpha$-entropy estimates and corresponding tightness arguments.
[04416] SOLUTIONS TO THE STOCHASTIC THIN-FILM EQUATION FOR INITIAL VALUES WITH NON-FULL SUPPORT
Format : Online Talk on Zoom
Author(s) :
Manuel Victor Gnann (Delft University of Technology)
Konstantinos Dareiotis (University of Leeds)
Benjamin Gess (Bielefeld University and Max Planck Institute for Mathematics in the Sciences, Leipzig)
Günther Grün (University of Erlangen-Nuremberg)
Max Sauerbrey (TU Delft)
Abstract : We prove existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilities including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone, and by Gruen, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise. The construction of solutions with non-full support for the initial data using alpha-entropies shall be discussed.
[04240] Existence of positive solutions to stochastic thin-film equations in the case of weak slippage
Format : Talk at Waseda University
Author(s) :
Lorenz Klein (FAU Erlangen-Nürnberg)
Günther Grün (FAU Erlangen-Nürnberg)
Abstract : We study stochastic thin-film equations for flow governed by surface tension and conjoining/disjoining interface potentials. For mobility exponents $n \in (2,3)$, we construct martingale solutions via spatial discretization, energy-entropy estimates based on stopping time arguments, and stochastic compactness methods. A crucial ingredient to extend methods used in the case $n=2$ are new discrete formulas for integration by parts which allow to treat nonlinearities related to Stratonovich correction terms.
This is joint work with G. Grün.
[04442] On finite speed of propagation for stochastic thin-film equations
Format : Online Talk on Zoom
Author(s) :
Günther Grün (FAU Erlangen-Nürnberg)
Lorenz Klein (FAU Erlangen-Nürnberg)
Abstract : In this talk, we present an energy method which allows to prove finite speed of propagation for sufficiently regular solutions to a class of stochastic thin-film equations with conservative multiplicative noise under periodic boundary conditions. Analytically, our approach is based on novel integral estimates combined with appropriate modifications of the technique previously used for stochastic porous-media and stochastic parabolic $p$-Laplace equations.
