- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 16:29:41
7
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=9&iss=1&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
9
2007
1
Noise regularization and computations for the 1-dimensional stochastic Allen–Cahn problem
Markos
Katsoulakis
University of Massachusetts, AMHERST, UNITED STATES
Georgios
Kossioris
Research Centre of Crete, VASILIKA VOUTON, GREECE
Omar
Lakkis
University of Sussex, BRIGHTON, UNITED KINGDOM
Allen-Cahn, Stochastic PDE, Finite Elements, Regularity, Mean Curvature Flow
We address the numerical discretization of the Allen-Cahn problem with additive white noise in one-dimensional space. Our main focus is to understand the behavior of the discretized equation with respect to a small ``interface thickness'' parameter and the noise intensity. The discretization is conducted in two stages: (1) regularize the white noise and study the regularized problem, (2) approximate the regularized problem. We address (1) by introducing a piecewise constant random approximation of the white noise with respect to a space-time mesh. We analyze the regularized problem and study its relation to both the original problem and the deterministic Allen-Cahn problem. Step (2) is then performed leading to a practical Monte-Carlo method combined with a Finite Element-Implicit Euler scheme. The resulting numerical scheme is tested against theoretical benchmark results concerning the behavior of the solution as the interface thickness goes to zero.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
1
30
10.4171/IFB/154
http://www.ems-ph.org/doi/10.4171/IFB/154
On generalized solutions of two-phase flows for viscous incompressible fluids
Helmut
Abels
Universität Regensburg, REGENSBURG, GERMANY
Two-phase flow, free boundary value problems, varifold solutions, measure-valued solutions, surface tension
We discuss the existence of generalized solutions of the flow of two immiscible, incompressible, viscous Newtonian and Non-Newtonian fluids with and without surface tension in a domain $\Omega\subseteq \R^d$, $d=2,3$. In the case without surface tension, the existence of weak solutions is shown, but little is known about the interface between both fluids. If surface tension is present, the energy estimates gives an a priori bound on the $(d-1)$-dimensional Hausdorff measure of the interface, but the existence of weak solutions is open. This might be due to possible oscillation and concentration effects of the interface related to instabilities of the interface as for example fingering, emulsification or just cancellation of area, when two parts of the interface meet. Nevertheless we will show the existence of so-called measure-valued varifold solutions, where the interface is modeled by an oriented general varifold $V(t)$ which is a non-negative measure on $\Omega\times \mathbb{S}^{d-1}$, where $\mathbb{S}^{d-1}$ is the unit sphere in $\R^d$. Moreover, it is shown that measure-valued varifold solutions are weak solution if an energy equality is satisfied.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
31
65
10.4171/IFB/155
http://www.ems-ph.org/doi/10.4171/IFB/155
A free boundary problem in glaciology: The motion of grounding lines
M.
Fontelos
Universidad Autónoma de Madrid, MADRID, SPAIN
Ana
Muñoz
Universidad Rey Juan Carlos I, MÓSTOLES (MADRID), SPAIN
Free boundary problem, Glaciology, mixed type boundary conditions, Stokes' flow problem, Lax-Milgram theorem, Mellin Transform
In this paper we consider stationary ice sheet modelled as a Stokes flow in a bounded two-dimensional domain. In particular, we study the behavior of the grounding line, where different boundary conditions meet: no-slip conditions for the grounded part and force balance conditions for the floating part whose shape is a priori undetermined. This yields a free boundary problem with mixed boundary conditions and a contact line, called "left grounding line" in the glaciological context, that might move along the solid substrate. We show that solutions with moving grounding lines and zero contact angle do exist and determine the shape and asymptotic properties of the free boundary.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
67
93
10.4171/IFB/156
http://www.ems-ph.org/doi/10.4171/IFB/156
An upper bound for the waiting time for doubly nonlinear parabolic equations
Kianhwa
Djie
RWTH Aachen, AACHEN, GERMANY
Nonlinear degenerate parabolic equations, waiting time
We obtain an upper bound for the waiting time for the doubly nonlinear parabolic equations $$ \left\{\begin{array}{rlll}(|u|^{q-2}u)_t - \textrm{div}(|\nabla u|^{p-2}\nabla u)&=&0&\textrm{in}\ \mathds{R}^N \times [0, \infty)\ ,\\ u(x, 0)&=&u_0(x)&\textrm{for all}\ x \in \mathds{R}^N\ ,\\ \end{array}\right.$$ depending on the growth of the initial value $u_0$ with parameters $p \geq 2,\ 1 < q < p$, and $|u_0|^{q-1} \in L^1(\mathds{R}^N)$. This upper bound coincides with the lower bound given by Giacomelli-Gr\"{u}n \cite{GG}. Therefore it is optimal. Special cases are the porous medium equation (for $p=2$) for which we obtain the result of Chipot-Sideris \cite{CS} and the parabolic $p$-Laplace equation (for $q=2$).
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
95
105
10.4171/IFB/157
http://www.ems-ph.org/doi/10.4171/IFB/157
Asymptotic analysis of Mumford–Shah type energies in periodically perforated domains
Matteo
Focardi
Università degli Studi di Firenze, FIRENZE, ITALY
Maria Stella
Gelli
Università di Pisa, PISA, ITALY
We study the asymptotic limit of obstacle problems for Mumford-Shah type functionals with $p$-growth in periodically-perforated domains \emph{via} the $\Gamma$-convergence of the associated free-discontinuity energies. In the limit a non-trivial penalization term related to the $1$-capacity of the reference hole appears if and only if the size of the perforation scales like $\eps^{\frac n{n-1}}$, being $\eps$ its periodicity. We give two different formulations of the obstacle problem to include also perforations with Lebesgue measure zero.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
107
132
10.4171/IFB/158
http://www.ems-ph.org/doi/10.4171/IFB/158
Uniqueness, symmetry and full regularity of free boundary in optimization problems with volume constraint
Eduardo
Teixeira
Universidade Federal do Ceará, FORTALEZA - CEARA, BRAZIL
In this paper we study qualitative geometric properties of optimal configurations to a variational problem with free boundary, under suitable assumptions on a fixed boundary. More specifically, we study the problem of minimizing the flow of heat given by $\int_{\partial D} \Gamma (u_\mu) d\sigma$, where $D$ is a fixed domain and $u$ is the potential of a domain $\Omega \supset \partial D$, with a prescribed volume on $\Omega \setminus D$. Our main goal is to establish uniqueness and symmetry results when $\partial D$ has a given geometric property. Full regularity of the free boundary is obtained under these symmetry conditions imposed on the fixed boundary.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
133
148
10.4171/IFB/159
http://www.ems-ph.org/doi/10.4171/IFB/159
Optimal channel networks, landscape function and branched transport
Filippo
Santambrogio
Scuola Normale Superiore, PISA, ITALY
Starting from transportation models for branching structures, we define a function that represents the elevation of the landscape in a river basin. This function is already well-known in the geophysical community but it is only considered under a very strong discretization. We generalize it to the continuous case and study its properties, providing several applications.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
149
169
10.4171/IFB/160
http://www.ems-ph.org/doi/10.4171/IFB/160