- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 12:59:22
7
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=7&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
7
2005
3
An algorithm for the elastic flow of surfaces
Raluca
Rusu
Universität Freiburg, FREIBURG I BR, GERMANY
A semi implicit fully discrete finite element scheme for the computation of the parametric elastic flow of two dimensional surfaces in $\r^3$ based on a variational form is presented. Linear finite elements are used for the space discretization of a mixed formulation. Time discretization is carried out by a semi implicit method to linearize the problem.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
229
239
10.4171/IFB/122
http://www.ems-ph.org/doi/10.4171/IFB/122
2-Dimensional flat curvature flow of crystals
David
Caraballo
Georgetown University, WASHINGTON, UNITED STATES
In the impressive and seminal paper \cite{ATW}, Fred Almgren, Jean Taylor, and Lihe Wang introduced flat curvature flow in $\mathbb{R}^{n}$, a variational time-discretization scheme for (isotropic or anisotropic) mean curvature flow. Their main result asserts the H\"{o}lder continuity, in time, of these flows. This essential estimate requires a boundary regularity result, a uniform lower density ratio bound condition, which they proved for each $n\geq 3.$ Similar estimates for Brownian flows, from important work by N.K Yip on stochastic mean curvature flow \cite{Yip}, also rely on this pivotal regularity result. In this paper, we complete these analyses for the case $n=2$ by establishing the necessary uniform lower density ratio bounds. MSC 2000 subject classification: 53C44, 49Q20, 49N60.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
241
254
10.4171/IFB/123
http://www.ems-ph.org/doi/10.4171/IFB/123
A system of degenerate parabolic nonlinear PDE's: a new free boundary problem
Michiel
Bertsch
Consiglio Nazionale delle Ricerche, ROMA, ITALY
Roberta
Dal Passo
Università di Roma, ROMA, ITALY
Carlo
Nitsch
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Nonlinear parabolic system, degenerate parabolic PDE's, nonlocal damage mechanics, oil engineering
We prove existence of solutions of a new free boundary problem described by a system of degenerate parabolic equations. The problem arises in petroleum engineering and concerns fluid flows in diatomite rocks. The unknown functions represent the pressure of the fluid and a damage parameter of the porous rock. These quantities are not necessarily continuous on the free boundary, which considerably complicates the mathematical analysis.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
255
276
10.4171/IFB/124
http://www.ems-ph.org/doi/10.4171/IFB/124
A coupled level set-boundary integral method for moving boundary simulations
M.
Garzon
Universidad de Oviedo, OVIEDO, SPAIN
David
Adalsteinsson
University of North Carolina at Chapel Hill, CHAPEL HILL, UNITED STATES
L.
Gray
Oak Ridge National Laboratory, Oak Ridge, UNITED STATES
James
Sethian
University of California, BERKELEY, UNITED STATES
A numerical method for moving boundary problems based upon Level Set and boundary integral formulations is presented. The interface velocity is obtained from the boundary integral solution using a Galerkin technique for post-processing function gradients on the interface. We introduce a new level set technique for propagating free boundary values in time, and couple this to a Narrow Band Level Set Method. Together, they allow us to both update the function values and the location of the interface. The methods are discussed in the context of the well-studied two-dimensional nonlinear potential flow model of breaking waves over a sloping beach. The numerical results show wave breaking and rollup, and the algorithm is verified by means of convergence studies and comparisons with previous techniques.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
277
302
10.4171/IFB/125
http://www.ems-ph.org/doi/10.4171/IFB/125
Geometric evolutions driven by threshold dynamics
Minsu
Song
University of Illinois at Urbana-Champaign, URBANA, UNITED STATES
Geometric evolution; threshold dynamics; viscosity solution
We study threshold dynamics on $\R^n$ which satisfies monotonicity, translation invariance and finite propagation speed. We develop the general schemes for the convergence of threshold dynamics to geometric evolutions governed by a velocity function depending on the normal direction alone.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
303
318
10.4171/IFB/126
http://www.ems-ph.org/doi/10.4171/IFB/126
The N-membranes problem for quasilinear degenerate systems
Assis
Azevedo
Universidade Do Minho, BRAGA, PORTUGAL
José Francisco
Rodrigues
FC Universidade de Lisboa, LISBOA, PORTUGAL
Lisa
Santos
Universidade do Minho, BRAGA, PORTUGAL
We study the regularity of the solution of the variational inequality for the problem of N-membranes in equilibrium with a degenerate operator of p-Laplacian type, 1
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
319
337
10.4171/IFB/127
http://www.ems-ph.org/doi/10.4171/IFB/127
Regularity of minimizers of quasi perimeters with a volume constraint
Qinglan
Xia
University of Texas at Austin, AUSTIN, UNITED STATES
In this article, we study the regularity of the boundary of sets minimizing a quasi perimeter $T\left( E\right) =P\left( E,\Omega \right) +G\left( E\right) $ with a volume constraint. Here $\Omega $ is any open subset of $ \mathbb{R}^{n}$ with $n\geq 2$, $G$ is a lower semicontinuous function on sets of finite perimeter satisfying a condition that $G\left( E\right) \leq G\left( F\right) +C\left| E\div F\right| ^{\beta }$ among all sets of finite perimeter with equal volume. We show that under the condition $\beta >1-\frac{1}{n}$, any volume constrained minimizer $E$ of the quasi perimeter $T$ has both interior points and exterior points, and $E$ is indeed a quasi minimizer of perimeter without the volume constraint. Using a well known regularity result about quasi minimizers of perimeter, we get the classical $ C^{1,\alpha }$ regularity for the reduced boundary of $E$.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
339
352
10.4171/IFB/128
http://www.ems-ph.org/doi/10.4171/IFB/128