Abstract : This minisymposium is focused on recent advances in the analysis of interface evolution problems. A particular
emphasis lies on prominent applications arising in materials science (grain coarsening in polycrystalline
materials), fluid mechanics (fluid-structure interaction, viscous surface waves, dynamic wetting) and phase
separation models from chemistry. The minisymposium brings together an international group of researchers,
new and established, to discuss topics covering a broad range of associated mathematical questions and
techniques. These include variational methods for modelling and solution theories, the rigorous derivation of
sharp interface limits, and the analysis of evolving networks of branched interfaces.
[02095] On the notion of generalized mean curvature flow
Format : Talk at Waseda University
Author(s) :
Yoshihiro Tonegawa (Tokyo Institute of Technology)
Abstract : I describe some new aspects of generalized notion of mean curvature flow which are prompted by various attempts to prove the existence, uniqueness and regularity. The main tool comes from geometric measure theory which allows treatments of singular geometric objects necessary to deal with the weak formulation of mean curvature flow.
[03166] Coarsening phenomena in the network flow
Format : Talk at Waseda University
Author(s) :
Alessandra Pluda (University of Pisa)
Abstract : A network is a 1-dimensional connected set in the plane composed of a finite number of curves that meet at their endpoints in junctions. The flow by curvature of networks has been introduced in mathematical materials science to model the evolution of polycrystals. We would like to formalise a coarsening behavior, suggested by numerical simulations. I will present the mathematical tools developed to describe this evolution and an argument which supports the coarsening behavior.
[03048] Uniqueness and stability of multiphase mean curvature flow beyond a singular time: the case of the shrinking circle
Format : Talk at Waseda University
Author(s) :
Alice Marveggio (Institute of Science and Technology Austria (ISTA))
Julian Fischer (Institute of Science and Technology Austria (ISTA))
Maximilian Moser (Institute of Science and Technology Austria (ISTA))
Sebastian Hensel (Hausdorff Center for Mathematics, University of Bonn)
Abstract : The evolution of a network of interfaces by mean curvature flow features the occurrence of topology changes and geometric singularities. As a consequence, classical solution concepts for mean curvature flow are in general limited to short-time existence theorems, which include singular times only for some stable shrinkers such as the circle. At the same time, the transition from strong to weak solution concepts (e.g. Brakke solutions) may lead to non-uniqueness of solutions.
Following the relative energy approach à la Fischer-Hensel-Laux-Simon and introducing a suitable notion of gradient-flow calibration for a shrinking circle, we prove a quantitative stability estimate holding up to the singular time. This implies a weak-strong uniqueness principle for weak BV solutions to planar multiphase mean curvature flow beyond a specific class of singularities.
Furthermore, we expect our method to have further applications to other types of shrinkers, as well as to prove quantitative convergence of diffuse-interface (Allen-Cahn) approximations for mean curvature flow.
[02872] Matrix-valued Allen–Cahn equation and the Keller–Rubinstein–Sternberg problem
Format : Talk at Waseda University
Author(s) :
Mingwen Fei
Fanghua Lin (Courant Institute of Mathematical Science, New York University)
Wei Wang (Zhejiang University)
Zhifei Zhang (Peking University)
Abstract : In this talk, we consider the sharp interface limit of a matrix-valued Allen-Cahn equation. We show that the sharp interface system is a two-phases flow system: the interface evolves according to the motion by mean curvature; in the two bulk phase regions, the solution obeys the heat flow of harmonic maps with values in $n\times n$ orthogonal matrices with determinant $+1$ and $-1$ respectively; on the interface, the phase matrices in two sides satisfy a novel mixed boundary condition. The above result provides a solution to the Keller-Rubinstein-Sternberg's problem in the $O(n)$ setting. Our proof relies on two key ingredients. First, in order to construct the approximate solutions by matched asymptotic expansions, as the standard approach does not seem to work, we introduce the notion of quasi-minimal connecting orbits. They satisfy the usual leading order equations up to some small higher order terms. In addition, the linearized systems around these quasi-minimal orbits needs to be solvable up to some good remainders. These flexibilities are needed for the possible ``degenerations" and higher dimensional kernels for the linearized operators on matrix-valued functions due to intriguing boundary conditions at the sharp interface. The second key point is to establish a spectral uniform lower bound estimate for the linearized operator around approximate solutions. To this end, we introduce additional decompositions to reduce the problem into the coercive estimates of several linearized operators for scalar functions and some singular product estimates which are accomplished by exploring special cancellation structures between eigenfunctions of these linearized operators.
[03054] Variational methods for time-dependent problems on dynamically changing domains
Format : Talk at Waseda University
Author(s) :
Malte Kampschulte (Charles University Prague)
Abstract : In this talk I will present a general, energetically consistent method that can be used to show the existence of weak solutions for nonlinear problems in fluid structure-interaction and related fields. Not only can this be done without the need to make simplifying assumptions on domain or equations, in fact it crucially relies on all physical terms being present. This talk is based on several recent results primarily with B.Benesova, S.Schwarzacher but also D.Breit, A.Cesik, G.Gravina and G.Sperone.
[04468] Sharp-interface limit of models with mechanics and contact lines
Format : Talk at Waseda University
Author(s) :
Dirk Peschka (WIAS Berlin & Freie Universität Berlin)
Leonie Schmeller (Weierstrass Institute)
Abstract : First, we construct gradient structures for free boundary problems including nonlinear elasticity, phase fields and moving contact lines, where the convergence of phase-field models to certain sharp-interface limits is analyzed numerically. Then, in the second part of the talk, it will be shown how shapes of droplets on soft elastic substrates can be predicted by corresponding (sharp-interface) models, and some emergent phenomena - cloaking and phase separation near the contact line - are pointed out.
[03766] Uniform Rectifiability for Minimizers of the Griffith Fracture Energy
Format : Talk at Waseda University
Author(s) :
Kerrek Stinson (University of Bonn)
Manuel Friedrich (University of Erlangen-Nuremberg)
Camille Labourie (University of Erlangen-Nuremberg)
Abstract : Recent studies for minimizers of the Griffith energy, which penalizes elastic energy and fracture, have relied on topological constraints for the (codimension-1) crack. Regularity results effectively say that if the crack locally separates the domain into different connected components, then the crack is in fact a smooth surface. Our analysis looks at minimizers without topological constraints. As a first step, we prove uniform rectifiability, which shows that on sufficiently small scales, the crack is nearly topologically separating. The purpose of this talk is to illuminate the new techniques applied in this setting and discuss how a similar approach can be used to prove a regularity result for the crack.
