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European Mathematical Society Publishing House
2024-03-28 18:27:53
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=10&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
10
2008
1
Nonfattening condition for the generalized evolution by mean curvature and applications
Samuel
Biton
Université de Tours, TOURS, FRANCE
Pierre
Cardaliaguet
Université de Bretagne Occidentale, BREST, FRANCE
Olivier
Ley
Université de Tours, TOURS, FRANCE
We prove a non fattening condition for a geometric evolution described by the level set approach. This condition is close to those of Soner \cite{soner93} and Barles, Soner and Souganidis \cite{bss93} but we apply it to some unbounded hypersurfaces. It allows us to prove uniqueness for the mean curvature equation for graphs with convex at infinity initial data, without any restriction on its growth at infinity, by seeing the evolution of the graph of a solution as a geometric motion.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
1
4
10.4171/IFB/177
http://www.ems-ph.org/doi/10.4171/IFB/177
Finite element approximation of a Cahn–Hilliard–Navier–Stokes system
David
Kay
University of Sussex, BRIGHTON, UNITED KINGDOM
Vanessa
Styles
University of Sussex, BRIGHTON, UNITED KINGDOM
Richard
Welford
University of Sussex, BRIGHTON, UNITED KINGDOM
We consider a semi-discrete and a practical fully-discrete finite element approximation of a Cahn - Hilliard - Navier - Stokes system. This system arises in the modelling of multiphase fluid systems. We show order $h$ error estimate between the solution of the system and the solution of the semi-discrete approximation. We also show the convergence of the fully discrete approximation. Finally, we present an efficient implementation of the fully discrete scheme together with some numerical simulations.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
15
43
10.4171/IFB/178
http://www.ems-ph.org/doi/10.4171/IFB/178
The Allen–Cahn action functional in higher dimensions
Luca
Mugnai
Mathematik in den Naturwissenschaften, LEIPZIG, GERMANY
Matthias
Röger
Max Planck Institute for Mathematics in the Scienc, LEIPZIG, GERMANY
Allen–Cahn equation, stochastic partial differential equations, large deviation theory, sharp interface limits, motion by mean curvature
The Allen--Cahn action functional is related to the probability of rare events in the stochastically perturbed Allen--Cahn equation. Formal calculations suggest a \emph{reduced action functional} in the sharp interface limit. We prove the corresponding lower bound in two and three space dimensions. One difficulty is that diffuse interfaces may collapse in the limit. We therefore consider the limit of diffuse surface area measures and introduce a generalized velocity and generalized reduced action functional in a class of evolving measures.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
45
78
10.4171/IFB/179
http://www.ems-ph.org/doi/10.4171/IFB/179
On the Lipschitz regularity of solutions of a minimum problem with free boundary
Aram
Karakhanyan
Australian National University, CANBERRA, AUSTRALIA
In this article under assumption of "small" density for negativity set, we prove local Lipschitz regularity for the one phase minimization problem with free boundary for the functional $$\mathcal E_p(v,\Omega)=\int_\Omega|\nabla v|^p+\lambda^p_1\X{u\leq0}+\lambda^p_2\X{u>0},\hspace{3mm} 1
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
79
86
10.4171/IFB/180
http://www.ems-ph.org/doi/10.4171/IFB/180
Optimal design under the one-dimensional wave equation
Faustino
Maestre
ETSI Industriales, CIUDAD REAL, SPAIN
Arnaud
Münch
Université de Franche-Comté, BESANCON CEDEX, FRANCE
Pablo
Pedregal
ETSI Industriales, CIUDAD REAL, SPAIN
An optimal design problem governed by the wave equation is examined in detail. Specifically, we seek the time-dependent optimal layout of two isotropic materials on a 1-d domain by minimizing a functional depending quadratically on the gradient of the state with coefficients that may depend on space, time and design. As it is typical in this kind of problems, they are ill-posed in the sense that there is not an optimal design. We therefore examine relaxation by using the representation of two-dimensional ($(x,t)\in\bkR^{2}$) divergence free vector fields as rotated gradients. By means of gradient Young measures, we transform the original optimal design problem into a non-convex vector variational problem, for which we can compute an explicit form of the ``\emph{constrained quasiconvexification }" of the cost density. Moreover, this quasiconvexification is recovered by first or second-order laminates which give us the optimal microstructure at every point. Finally, we analyze the relaxed problem and some numerical experiments are performed. The perspective is similar to the one developed in previous papers for linear elliptic state equations. The novelty here lies in the state equation (the wave equation), and our contribution consists in understanding the differences with respect to elliptic cases.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
87
117
10.4171/IFB/181
http://www.ems-ph.org/doi/10.4171/IFB/181
Eulerian finite element method for parabolic PDEs on implicit surfaces
Gerhard
Dziuk
Universität Freiburg, FREIBURG I BR, GERMANY
Charles
Elliott
University of Warwick, COVENTRY, UNITED KINGDOM
We define an Eulerian level set method for parabolic partial differential equations on a stationary hypersurface $\Gamma$ contained in a domain $\Omega \subset \mathbb R^{n+1}$. The method is based on formulating the partial differential equations on all level surfaces of a prescribed function $\Phi$ whose zero level set is $\Gamma$. Eulerian surface gradients are formulated by using a projection of the gradient in $\mathbb R^{n+1}$ onto the level surfaces of $\Phi$. These Eulerian surface gradients are used to define weak forms of surface elliptic operators and so generate weak formulations of surface elliptic and parabolic equations. The resulting equation is then solved in one dimension higher but can be solved on a mesh which is unaligned to the level sets of $\Phi$. We consider both second order and fourth order elliptic operators with natural second order splittings. The finite element method is applied to the weak form of the split system of second order equations using piece-wise linear elements on a fixed grid. The computation of the mass and element stiffness matrices are simple and straightforward. Numerical experiments are described which indicate the power of the method. We describe how this framework may be employed in applications.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
119
138
10.4171/IFB/182
http://www.ems-ph.org/doi/10.4171/IFB/182
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
3
Threshold dynamics for high order geometric motions
Selim
Esedoglu
University of Michigan, ANN ARBOR, UNITED STATES
Steven
Ruuth
Simon Fraser University, BURNABY, BC, CANADA
Richard
Tsai
University of Texas at Austin, AUSTIN, UNITED STATES
In this paper, a class of algorithms for the high order geometric motion of planar curves is developed. The algorithms alternate two simple steps—a convolution and a thresholding step—to evolve planar curves according to combinations of Willmore flow, surface diffusion flow and curvature motion. A distinguishing feature of the methods is that they posses much better stability than typical explicit algorithms. Formal expansions and numerical examples are provided for a variety of high order flows to validate the methods and illustrate their behaviors.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
263
282
10.4171/IFB/189
http://www.ems-ph.org/doi/10.4171/IFB/189
Implicit time discretization of the mean curvature flow with a discontinuous forcing term
Antonin
Chambolle
Ecole Polytechnique, CNRS, PALAISEAU, FRANCE
Matteo
Novaga
Università di Pisa, PISA, ITALY
We consider an implicit time discretization for the motion of a hypersurface driven by its anisotropic mean curvature. We prove some convergence results of the scheme under very general assumptions on the forcing term, which include in particular the case of a typical path of the Brownian motion. We compare this limit with other available solutions, whenever they are defined. As a by-product of the analysis, we also provide a simple proof of the coincidence of the limit flow with the regular evolutions, defined for small times, in the case of a regular forcing term.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
283
300
10.4171/IFB/190
http://www.ems-ph.org/doi/10.4171/IFB/190
The critical mass constraint in the Cahn–Hilliard equation
Xiaofeng
Ren
Utah State University, LOGAN, UNITED STATES
Juncheng
Wei
University of British Columbia, VANCOUVER, CANADA
When the mass constraint of the Cahn-Hilliard equation in two dimensions is lowered to the order of $\e^{2/3}$, where $\e$ is the interface thickness parameter, the existence of droplet solutions becomes conditional. For interior single droplet solutions, there is a critical value for the mass constraint such that above this value two interior single droplet solutions exist, and below this value interior single droplet solutions can not be constructed. One solution has smaller droplet radius than the other one does. The one with smaller radius is less stable than the one with larger radius. The center of the droplets in these solutions is (almost) the point in the domain that is furthest from the boundary. A critical mass constraint also appears when multiple droplet solutions are sought. Above the critical mass constraint, which now depends on the number of droplets, there exist two multi-droplet solutions. In each solution the radii of the droplets are about the same. However when the two solutions are compared, one has larger droplet radius than the other one does. The locations of the droplets are determined by the solution of a disc packing problem.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
301
338
10.4171/IFB/191
http://www.ems-ph.org/doi/10.4171/IFB/191
Viscosity solutions of discontinuous Hamilton–Jacobi equations
Xinfu
Chen
University of Pittsburgh, PITTSBURGH, UNITED STATES
Bei
Hu
University of Notre Dame, NOTRE DAME, UNITED STATES
We define viscosity solutions for the Hamilton-Jacobi equation $ \phi_t = v(x,t) H(\nabla \phi)$ in $\BR^N\times(0,\infty)$ where $v$ is positive and bounded measurable and $H$ is non-negative and Lipschitz continuous. Under certain assumptions, we establish the existence and uniqueness of Lipschitz continuous viscosity solutions. The uniqueness result holds in particular for those $v$ which are independent of $t$ and piecewise continuous with discontinuous sets consisting of finitely many smooth lower dimensional surfaces not tangent to each other at any point of their intersection.ets are determined by the solution of a disc packing problem.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
339
359
10.4171/IFB/192
http://www.ems-ph.org/doi/10.4171/IFB/192
On a free boundary problem modelling inductive-heating processes
Hong-Ming
Yin
Washington State University, PULLMAN, UNITED STATES
Free boundary problem, induction heating, superheating
In this paper we study a free boundary problem describing a melting process by using induction heating. The mathematical model in one-space dimension consists of a coupled-parabolic system in each phase along with a non-equilibrium kinetic condition on the interface. By applying an energy estimate and a Campanato type of estimates, it is shown that the problem has a unique classical solution globally.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
361
375
10.4171/IFB/193
http://www.ems-ph.org/doi/10.4171/IFB/193
Traveling wave solutions of a nonlinear degenerate parabolic system from petroleum engineering
Michiel
Bertsch
Consiglio Nazionale delle Ricerche, ROMA, ITALY
Carlo
Nitsch
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
We study existence and qualitative properties of traveling wave solutions of a new free boundary problem which describes fluid flow in diatomite rocks. Diatomites are rather fragile and are characterized by a low permeability, which can increase due to the nonlocal accumulation of damage caused by the fluid flow. The traveling wave solutions give insight in the behavior near the free boundaries and show a strong parameter dependence. In particular we find in certain parameter ranges solutions with discontinuities across the free boundaries.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
377
398
10.4171/IFB/194
http://www.ems-ph.org/doi/10.4171/IFB/194
Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I
Benjamin
Boutin
Université Pierre et Marie Curie, PARIS, FRANCE
Christophe
Chalons
Université Pierre et Marie Curie-Paris 6, PARIS, FRANCE
Frédéric
Lagoutière
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Philippe
LeFloch
Université Pierre et Marie Curie, PARIS, FRANCE
We propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to standard finite difference schemes. The main challenge is to achieve, at the discretization level, a consistency property with respect to a prescribed kinetic relation. The latter is required for the selection of physically meaningful nonclassical shocks. Our method is based on a reconstruction technique performed in each computational cell that may contain a nonclassical shock. To validate this approach, we establish several consistency and stability properties, and we perform careful numerical experiments. The convergence of the algorithm toward the physically meaningful solutions selected by a kinetic relation is demonstrated numerically for several test cases, including concave-convex as well as convex-concave flux-functions.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
399
421
10.4171/IFB/195
http://www.ems-ph.org/doi/10.4171/IFB/195
4
Properties of Sobolev-type metrics in the space of curves
A.C.G.
Mennucci
Scuola Normale Superiore, PISA, ITALY
A.
Yezzi
Georgia Institute of Technology, ATLANTA, UNITED STATES
G.
Sundaramoorthi
Georgia Institute of Technology, ATLANTA, UNITED STATES
We define a manifold $M$ where objects $c\in M$ are curves, which we parameterize as $c:S^1\to \real^n$ ($n\ge 2$, $S^1$ is the circle). We study geometries on the manifold of curves, provided by Sobolev--type Riemannian metrics $H^j$. These metrics have been shown to regularize gradient flows used in Computer Vision applications, see \cite{ganesh:SAC07, ganesh:new_sobol_activ_contour08} and references therein. We provide some basic results of $H^j$ metrics; and, for the cases $j=1,2$, we characterize the completion of the space of smooth curves. We call these completions \emph{``$H^1$ and $H^2$ Sobolev--type Riemannian Manifolds of Curves''}. \gsinsert{This result is fundamental since it is a first step in proving the existence of geodesics with respect to these metrics.} As a byproduct, we prove that the Fr\'echet distance of curves (see \cite{Michor-Mumford}) coincides with the distance induced by the ``Finsler $L^\infinity$ metric'' defined in \S2.2 in \cite{YM:metrics04}
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
423
445
10.4171/IFB/196
http://www.ems-ph.org/doi/10.4171/IFB/196
A quasilinear parabolic singular perturbation problem
Claudia
Lederman
Universidad de Buenos Aires, BUENOS AIRES, ARGENTINA
Dietmar
Oelz
Universität Wien, WIEN, AUSTRIA
Quasilinear parabolic operator, singular perturbation problem, free boundary problem, combustion
We study a singular perturbation problem for a quasilinear uniformly parabolic operator of interest in combustion theory. We obtain uniform estimates, we pass to the limit and we show that, under suitable assumptions, the limit function $u$ is a solution to the free boundary problem ${\rm div } F(\nabla u)-\partial_{t}u=0$ in $\{ u>0 \}$, $u_\nu=\alpha(\nu, M)$ on $\partial\{ u>0 \}$, in a pointwise sense and in a viscosity sense. Here $\nu$ is the inward unit spatial normal to the free boundary $\partial\{ u>0 \}$ and $M$ is a positive constant. Some of the results obtained are new even when the operator under consideration is linear.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
447
482
10.4171/IFB/197
http://www.ems-ph.org/doi/10.4171/IFB/197
Long-time asymptotics of Hele–Shaw flow for perturbed balls with injection and suction
E.
Vondenhoff
TU Eindhoven, EINDHOVEN, NETHERLANDS
Hele–Shaw flow, linearised stability, abstract quasilinear parabolic evolution equations
We discuss long-time behaviour of Hele-Shaw flow with injection and suction, for domains that are small perturbations of balls. An evolution equation for the motion of these domains is derived and linearised. We use spectral properties of the linearisation to show that in the case of injection, perturbations of balls decay algebraically. For classical Hele-Shaw flow, convergence turns out to be faster if low Richardson moments vanish. If for the three-dimensional case surface tension is included, all liquid can be removed by suction if the suction point and the geometric centre coincide and the ratio of suction speed and surface tension is small enough. An arbitrarily large portion of the liquid can be removed if the initial domain is sufficiently close to a ball. The main tools are the principle of linearised stability and the theory of H. Amann for abstract quasilinear parabolic evolution equations.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
483
502
10.4171/IFB/198
http://www.ems-ph.org/doi/10.4171/IFB/198
Variational models for peeling problems
Francesco
Maddalena
Università degli Studi di Bari, BARI, ITALY
Danilo
Percivale
Università di Genova, GENOVA, ITALY
We study variational models for flexural beams and plates interacting with a rigid substrate through an adhesive layer. The general structure of the minimizers is investigated and some properties characterizing the system's dependence from the load and the material stiffnesses are discussed.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
503
516
10.4171/IFB/199
http://www.ems-ph.org/doi/10.4171/IFB/199
Traveling waves for the Keller–Segel system with Fisher birth terms
Gregoire
Nadin
Ecole Normale Superieure, PARIS CEDEX 05, FRANCE
Benoît
Perthame
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Lenya
Ryzhik
Stanford University, STANFORD, UNITED STATES
Chemotaxis, traveling waves, Keller–Segel system, reaction-diffusion systems, nonlinear stability
We consider the traveling wave problem for the one dimensional Keller-Segel system with a birth term of either a Fisher/KPP type or with a truncation for small population densities. We prove that there exists a solution under some stability conditions on the coefficients which enforce an upper bound on the solution and $\dot H^1(\R)$ estimates. Solutions in the KPP case are built as a limit of traveling waves for the truncated birth rates (similar to ignition temperature in combustion theory). We also discuss some general bounds and long time convergence for the solution of the Cauchy problem and in particular linear and nonlinear stability of the non-zero steady state.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
517
538
10.4171/IFB/200
http://www.ems-ph.org/doi/10.4171/IFB/200
Anisotropic mean curvature flow for two-dimensional surfaces in higher codimension: a numerical scheme
Paola
Pozzi
Universität Duisburg-Essen, ESSEN, GERMANY
We consider evolution of two dimensional parametric surfaces by ani\-sotropic mean curvature flow in $\R^n$, for arbitrary $n \geq 3$. After deriving a classical and a weak formulation of the flow, a fully discrete stable finite element scheme is proposed, and numerical tests and simulations are presented.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
539
576
10.4171/IFB/201
http://www.ems-ph.org/doi/10.4171/IFB/201