Dresden University of Technology

Stochastic homogenization shows that solutions to an elliptic problem 

with rapidly oscillating, ergodic random coefficients can be effectively 

described by an elliptic problem with homogeneous, deterministic 

coefficients. The definition of the latter is based on the construction 

of a "corrector" and invokes an elliptic operator that acts on the 

probability space of admissible coefficient fields. While qualitative 

homogenization is well understood and classical, quantitative results 

(e.g. estimates on the homogenization error and approximations to the 

homogenized coefficients) have only been obtained recently.  In the talk 

we discuss an optimal estimate on the associated semigroup (usually 

called the "random walk in the random environment") and show that it 

decays with an algebraic rate. The result relies on a link between a 

Spectral Gap of a Glauber dynamics on the space of coefficient fields (a 

notion that we borrow from statistical mechanics) and heat kernel 

estimates. As applications we obtain moment bounds on the corrector and 

an optimal convergence rate for the approximation of the homogenized 

coefficients via periodic representative volume elements.

Motivated by stochastic models for order books in stock exchanges we consider stochastic partial differential equations with a free boundary condition. Such equations can be considered generalizations of the classic (deterministic) Stefan problem of heat condition in a two-phase medium. 

Extending results by Kim, Zheng & Sowers we allow for non-linear boundary interaction, general Robin-type boundary conditions and fairly general drift and diffusion coefficients. Existence of maximal local and global solutions is established by transforming the equation to a fixed-boundary problem and solving a stochastic evolution equation in suitable interpolation spaces. Based on joint work with Marvin Mueller.

Motivated by stochastic models for order books in stock exchanges we consider stochastic partial differential equations with a free boundary condition. Such equations can be considered generalizations of the classic (deterministic) Stefan problem of heat condition in a two-phase medium. 

Extending results by Kim, Zheng & Sowers we allow for non-linear boundary interaction, general Robin-type boundary conditions and fairly general drift and diffusion coefficients. Existence of maximal local and global solutions is established by transforming the equation to a fixed-boundary problem and solving a stochastic evolution equation in suitable interpolation spaces. Based on joint work with Marvin Mueller.

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Starting from a Navier-Stokes-Cahn-Hilliard equation for a two-phase flow problem we discuss efficient numerical approaches based on adaptive finite element methods. Various extensions of the model are discussed: a) we consider the model on implicitly described geometries, which is used to simulate the sliding of droplets over nano-patterned surfaces, b) we consider the effect of soluble surfactants and show its influence on tip splitting of droplets under shear flow, and c) we consider bijels as a new class of soft matter materials, in which colloidal particles are jammed on the fluid-fluid interface and effect the motion of the interface due to an elastic force.

The work is based on joint work with Sebastian Aland (TU Dresden), John Lowengrub (UC Irvine) and Knut Erik Teigen (U Trondheim).