- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 14:47:40
6
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=14&iss=2&update_since=2024-03-29
Interfaces and Free Boundaries
Interfaces Free Bound.
IFB
1463-9963
1463-9971
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
10.4171/IFB
http://www.ems-ph.org/doi/10.4171/IFB
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
14
2012
2
Regular solutions to a monodimensional model with discontinuous elliptic operator
Piotr
Mucha
University of Warsaw, WARSAW, POLAND
Singular elliptic operator, discontinuity, facets, qualitative analysis, structure of solutions
The note examines qualitative behavior of solutions to a monodimensional nonlinear elliptic equation $\frac{d}{dx}(u_x+\mathrm {sgn} \ u_x)=f$ with Dirichlet boundary data. This simple example explains the phenomenon of facets – flat regions of solutions, characteristic for models arising from theories of crystal growth and image prossessing.
Partial differential equations
Mechanics of deformable solids
Information and communication, circuits
General
145
152
10.4171/IFB/276
http://www.ems-ph.org/doi/10.4171/IFB/276
Density estimates for phase transitions with a trace
Yannick
Sire
Université Aix-Marseille, MARSEILLE CEDEX 13, FRANCE
Enrico
Valdinoci
Università degli Studi di Milano, MILANO, ITALY
Singular and degenerate boundary reaction equations, measure theoretic estimates of the level lets of phase transition layers
We consider a functional obtained by adding a trace term to the Allen-Cahn phase segregation model and we prove some density estimates for the level sets of the interfaces. We treat in a unified way also the cases of possible degeneracy and singularity of the ellipticity of the model and the quasiminimal case.
Partial differential equations
Statistical mechanics, structure of matter
General
153
165
10.4171/IFB/277
http://www.ems-ph.org/doi/10.4171/IFB/277
Capillary drops on a rough surface
Antoine
Mellet
University of Maryland, COLLEGE PARK, UNITED STATES
James
Nolen
Duke University, DURHAM, UNITED STATES
Capillary drops, homogenization, random media, free boundary problems
We study liquid drops lying on a rough planar surface. The drops are minimizers of an energy functional that includes a random adhesion energy. We prove the existence of minimizers and the regularity of the free boundary. When the length scale of the randomly varying surface is small, we show that minimizers are close to spherical caps which are minimizers of an averaged energy functional. In particular, we give an error estimate that is algebraic in the scale parameter and holds with high probability.
Fluid mechanics
Partial differential equations
General
167
184
10.4171/IFB/278
http://www.ems-ph.org/doi/10.4171/IFB/278
Two-phase flow problem coupled with mean curvature flow
Chun
Liu
The Pennsylvania State University, UNIVERSITY PARK, UNITED STATES
Norifumi
Sato
Furano H.S., FURANO (HOKKAIDO), JAPAN
Yoshihiro
Tonegawa
Hokkaido University, SAPPORO, JAPAN
Two-phase fluid, surface energy, varifold, phase field method
We prove the existence of generalized solution for incompressible and viscous non-Newtonian two-phase fluid flow for spatial dimension $d =2$ and 3. Separating two shear thickening fluids with power law viscosity strictly above critical growth $p = (d + 2)/2$, the phase boundary moves along with the fluid flow plus its mean curvature while exerting surface tension force to the fluid. An approximation scheme combining the Galerkin method and the phase field method is adopted.
Partial differential equations
Fluid mechanics
General
185
203
10.4171/IFB/279
http://www.ems-ph.org/doi/10.4171/IFB/279
Hele–Shaw flow in thin threads: A rigorous limit result
Bogdan-Vasile
Matioc
Leibniz University Hannover, HANNOVER, GERMANY
Georg
Prokert
TU Eindhoven, EINDHOVEN, NETHERLANDS
Hele–Shaw flow, surface tension, Thin Film equation, degenerate parabolic equation
We rigorously prove the convergence of appropriately scaled solutions of the 2D Hele–Shaw moving boundary problem with surface tension in the limit of thin threads to the solution of the formally corresponding Thin Film equation. The proof is based on scaled parabolic estimates for the nonlocal, nonlinear evolution equations that arise from these problems.
Partial differential equations
Fluid mechanics
General
205
230
10.4171/IFB/280
http://www.ems-ph.org/doi/10.4171/IFB/280
Finite element methods for director fields on flexible surfaces
Sören
Bartels
Universität Bonn, BONN, GERMANY
Georg
Dolzmann
Universität Regensburg, REGENSBURG, GERMANY
Ricardo
Nochetto
University of Maryland, COLLEGE PARK, UNITED STATES
Alexander
Raisch
Universität Bonn, BONN, GERMANY
Finite elements, surfaces, biomembranes, surfactants, director fields
We introduce a nonlinear model for the evolution of biomembranes driven by the $L^2$-gradient flow of a novel elasticity functional describing the interaction of a director field on a membrane with its curvature. In the linearized setting of a graph we present a practical finite element method (FEM), and prove a priori estimates. We derive the relaxation dynamics for the nonlinear model on closed surfaces and introduce a parametric FEM. We present numerical experiments for both linear and nonlinear models, which agree well with the expected behavior in simple situations and allow predictions beyond theory.
Partial differential equations
Mechanics of deformable solids
General
231
272
10.4171/IFB/281
http://www.ems-ph.org/doi/10.4171/IFB/281