- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 00:16:27
6
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=10&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
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
10
2008
2
A convolution thresholding scheme for the Willmore flow
Richards
Grzhibovskis
Universität des Saarlandes, SAARBRÜCKEN, GERMANY
Alexei
Heintz
Chalmers University of Technology, GOTHENBURG, SWEDEN
Willmore flow, convolution thresholding scheme
A convolution thresholding approximation scheme for the Willmore geometric flow is constructed. It is based on an asymptotic expansion of the convolution of an indicator function with a smooth, isotropic kernel. The consistency of the method is justified when the evolving surface is smooth and embedded. Some aspects of the numerical implementation of the scheme are discussed and several numerical results are presented. Numerical experiments show that the method performs well even in the case of a non-smooth initial data.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
139
153
10.4171/IFB/183
http://www.ems-ph.org/doi/10.4171/IFB/183
Higher dimensional problems with volume constraints—Existence and Γ-convergence
Marc Oliver
Rieger
University of Zürich, ZÜRICH, SWITZERLAND
We study variational problems with volume constraints (also called level set constraints) of the form \begin{eqnarray*} \mbox{Minimize }E(u):=\int_\G f(u,\nabla u)\,dx,\nonumber\\ |\{x\in\Omega,\;u(x)=a\}|=\alpha,\quad |\{x\in\Omega,\;u(x)=b\}|=\beta, \end{eqnarray*} on $\Omega\subset\R^n$, where $u\in H^1(\G)$ and $\alpha+\beta
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
155
172
10.4171/IFB/184
http://www.ems-ph.org/doi/10.4171/IFB/184
Coexistence and segregation for strongly competing species in special domains
Monica
Conti
Politecnico, MILANO, ITALY
Veronica
Felli
Università degli Studi di Milano-Bicocca, MILANO, ITALY
We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Dirichlet boundary conditions. For a class of nonconvex domains composed by balls connected with thin corridors, we show the occurrence of pattern formation (coexistence and spatial segregation of all the species), as the competition grows indefinitely. As a result we prove the existence and uniqueness of solutions for a remarkable system of differential inequalities involved in segregation phenomena and optimal partition problems.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
173
195
10.4171/IFB/185
http://www.ems-ph.org/doi/10.4171/IFB/185
Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling
Christophe
Chalons
Université Pierre et Marie Curie-Paris 6, PARIS, FRANCE
Paola
Goatin
Université du Sud Toulon - Var, LA VALETTE DU VAR CEDEX, FRANCE
Hyperbolic conservation laws, continuous traffic models, Godunov scheme, phase transitions, sampling
A new version of Godunov scheme is proposed in order to compute solutions of a traffic flow model with phase transitions. The scheme is based on a modified averaging strategy and a sampling procedure. Several numerical tests are shown to prove the validity of the method. The convergence of the algorithm is demonstrated numerically. We also give a higher order extension of the method in space and time.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
197
221
10.4171/IFB/186
http://www.ems-ph.org/doi/10.4171/IFB/186
On the energy of a flow arising in shape optimization
Pierre
Cardaliaguet
Université de Bretagne Occidentale, BREST, FRANCE
Olivier
Ley
Université de Tours, TOURS, FRANCE
In \cite{cl05} we have defined a viscosity solution for the gradient flow of the exterior Bernoulli free boundary problem. We prove here that the associated energy is non increasing along the flow. For this we build a discrete gradient flow in the flavour of Almgren, Taylor and Wang \cite{atw93}.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
223
243
10.4171/IFB/187
http://www.ems-ph.org/doi/10.4171/IFB/187
A multiscale tumor model
Avner
Friedman
Ohio State University, COLUMBUS, UNITED STATES
We consider a tumor model with two time scales: the time $t$ during which the tumor evolves and the running time $s_{i}$ for each of the phases of the cell cycle for the cells in the tumor. The model also includes the effect of genes mutations in the sense that populations of cells with different mutations and in different phases of the cell cycle evolve by different rules. The model is formulated as a coupled system of partial differential equations; a transition from one population to another occurs at the `restriction points' located at the ends of the $G_{1}$ and $S$ phases. The PDEs for the cell populations are hyperbolic equations based on mass conservation laws. The model includes also a diffusion equation for the oxygen concentration and an elliptic equation for the internal pressure caused by proliferation and death of cells. The tumor region is viewed as a domain with a moving boundary, satisfying a continuity equation at the free boundary. Existence and uniqueness are proved for a small time interval, for general initial conditions, and for all time in the case of radially symmetric initial conditions.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
245
262
10.4171/IFB/188
http://www.ems-ph.org/doi/10.4171/IFB/188