- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 11:41:35
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=14&iss=3&update_since=2024-03-28
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
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
14
2012
3
Existence of strong solutions for the motion of an elastic structure in an incompressible viscous fluid
Muriel
Boulakia
Université Pierre et Marie Curie, PARIS, FRANCE
Erica
Schwindt
Universidad de Chile, SANTIAGO, CHILE
Takéo
Takahashi
Université Henri Poincaré, VANDOEUVRE-LES-NANCY, FRANCE
Fluid-structure interaction, existence and uniqueness of strong solutions, incompressible Navier–Stokes equations, deformable structure
In this paper we study a three-dimensional fluid–structure interaction problem. The motion of the fluid is modeled by the Navier–Stokes equations and we consider for the elastic structure a finite dimensional approximation of the equation of linear elasticity. The time variation of the fluid domain is not known a priori, so we deal with a free boundary value problem. Our main result yields the local in time existence and uniqueness of strong solutions for this system.
Mechanics of deformable solids
Partial differential equations
Dynamical systems and ergodic theory
Fluid mechanics
273
306
10.4171/IFB/282
http://www.ems-ph.org/doi/10.4171/IFB/282
A two-phase problem with a lower-dimensional free boundary
Mark
Allen
Purdue University, WEST LAFAYETTE, UNITED STATES
Arshak
Petrosyan
Purdue University, WEST LAFAYETTE, UNITED STATES
Two-phase free boundary problem, lower-dimensional free boundary, separation of phases, regularity of the free boundary, monotonicity formula, Alexandrov reflection technique, Steiner symmetrization
For a bounded domain $D\subset \R^n$, we study minimizers of the energy functional \[ \int_{D}{|\nabla u|^2}\,dx + \int_{D \cap (\R^{n-1} \times \{0\} )}{\lambda^+ \chi_{ \{u > 0\} } + \lambda^- \chi_{ \{u 0\}\quad \text{and}\quad \Gamma^- = \partial \{u(\cdot, 0) < 0\} \] never touch. Moreover, using Alexandrov-type reflection technique, we can show that in dimension $n=3$ the free boundaries are $C^1$ regular on a dense subset.
Partial differential equations
General
307
342
10.4171/IFB/283
http://www.ems-ph.org/doi/10.4171/IFB/283
Existence and approximation of a nonlinear degenerate parabolic system modelling acid-mediated tumour invasion
John
Barrett
Imperial College London, LONDON, UNITED KINGDOM
Klaus
Deckelnick
Otto-von-Guericke-Universität Magdeburg, MAGDEBURG, GERMANY
Degenerate parabolic system, porous medium equation, existence, finite elements, tumour invasion
We consider a nonlinear parabolic system of reaction–diffusion equations modelling acid-mediated tumour invasion. The system couples potentially degenerate equations for the cell densities of the normal and tumour populations to a parabolic equation for the concentration of HC ions. We obtain an existence result for the system by constructing a suitable finite element approximation and analyzing its convergence. Finally, we report on corresponding numerical experiments.
Partial differential equations
Numerical analysis
General
343
363
10.4171/IFB/284
http://www.ems-ph.org/doi/10.4171/IFB/284
Uniqueness and existence of spirals moving by forced mean curvature motion
Nicolas
Forcadel
Université Paris-Dauphine, PARIS CEDEX 16, FRANCE
Cyril
Imbert
Université Paris-Est Créteil Val de Marne, CRÉTEIL CEDEX, FRANCE
Régis
Monneau
Cité Descartes - Champs sur Marne, MARNE-LA-VALLÉE CEDEX 2, FRANCE
Spirals, motion of interfaces, comparison principle, quasi-linear parabolic equation, viscosity solutions, mean curvature motion
In this paper, we study the motion of spirals by mean curvature type motion in 1 the (two dimensional) plane. Our motivation comes from dislocation dynamics; in this context, spirals appear when a screw dislocation line reaches the surface of a crystal. The first main result of this paper is a comparison principle for the corresponding parabolic quasi-linear equation. As far as motion of spirals are concerned, the novelty and originality of our setting and results come from the fact that, first, the singularity generated by the attached end point of spirals is taken into account for the first time, and second, spirals are studied in the whole space. Our second main result states that the Cauchy problem is well-posed in the class of sub-linear weak (viscosity) solutions. We also explain how to get the existence of smooth solutions when initial data satisfy an additional compatibility condition.
Partial differential equations
General
365
400
10.4171/IFB/285
http://www.ems-ph.org/doi/10.4171/IFB/285
Topology optimization methods with gradient-free perimeter approximation
Samuel
Amstutz
Université d'Avignon, AVIGNON, FRANCE
Nicolas
Van Goethem
Universidade de Lisboa, LISBOA, PORTUGAL
Topology optimization, perimeter, $\Gamma$-convergence, homogenization
In this paper we introduce a family of smooth perimeter approximating functionals designed to be incorporated within topology optimization algorithms. The required mathematical properties, namely the $\Gamma$-convergence and the compactness of sequences of minimizers, are first established. Then we propose several methods for the solution of topology optimization problems with perimeter penalization showing different features. We conclude by some numerical illustrations in the contexts of least square problems and compliance minimization.
Calculus of variations and optimal control; optimization
Partial differential equations
General
401
430
10.4171/IFB/286
http://www.ems-ph.org/doi/10.4171/IFB/286