- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 16:29:21
19
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=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
2
2000
1
Existence and uniqueness of a classical solution for a mathematical model describing the isobaric crystallization of a polymer
Antonio
Fasano
Universita di Firenze, FIRENZE, ITALY
Alberto
Mancini
Università degli Studi di Firenze, FIRENZE, ITALY
Free boundary problems; nonlinear parabolic equations; phase change
In this paper a global existence and uniqueness result is presented for the classical solution of a free boundary problem for a system of partial differential equations (p.d.e.s.) with non-local boundary conditions describing the crystallization process of a cylindrical sample of polymer under prescribed pressure. The system of equations is discussed in [16] as the model for coupled cooling and shrinking of a sample of molten polymer under a given constant pressure. The velocity field generated by the thermal and chemical contraction enters the model only through its divergence. Such an approximation is discussed on the basis of a qualitative analysis.
Functional analysis
Probability theory and stochastic processes
1
19
10.4171/IFB/11
http://www.ems-ph.org/doi/10.4171/IFB/11
A free boundary problem involving a cusp: breakthrough of salt water
Hans Wilhelm
Alt
Universität Bonn, BONN, GERMANY
C
van Duijn
Centre for Mathematics and Computer Science, AMSTERDAM, NETHERLANDS
Porous media flow; free boundary problem
In this paper we study a two-phase free boundary problem describing the stationary flow of fresh and salt water in a porous medium, when both fluids are drawn into a well. For given discharges at the well (Qf for fresh water and Qs for salt water) we formulate the problem in terms of the stream function in an axial symmetric flow domain in Rn(n=2,3). We prove the existence of a continuous free boundary which ends up in the well, located on the central axis. Moreover, we show that the free boundary has a tangent at the well and approaches it in a C1 sense. Using the method of separation of variables we also give a result concerning the asymptotic behaviour of the free boundary at the well. For a given total discharge Q:=Qf + Qs we consider the vanishing Qs limit. We show that a free boundary arises with a cusp at the central axis, having a positive distance from the well. This work is a continuation of [5,6].
Functional analysis
Probability theory and stochastic processes
21
72
10.4171/IFB/12
http://www.ems-ph.org/doi/10.4171/IFB/12
Phenomenological modelling of polymer crystallization using the notion of multiple natural configurations
I.
Rao
Texas A&M University, COLLEGE STATION, UNITED STATES
K.
Rajagopal
Texas A&M University, COLLEGE STATION, UNITED STATES
polymer crystallization, multiple natural configurations, homogenous deformations
Crystallization and solidification in polymers is a problem of great importance to the polymer processing industry. In these processes, the melt is subjected to deformation while being cooled into the desired shape. The properties of the final product are strongly influenced by the deformation and thermal histories and the final solid is invariably anisotropic. In this work we present a model to capture the effects during solidification and crystallization in polymers within a purely mechanical setting, using the framework of multiple natural configurations that was introduced recently to study a variety of non-linear dissipative responses of materials undergoing phase transitions. Using this framework we present a consistent method to model the transition from a fluid-like behaviour to a solid-like behaviour. We also present a novel way of incorporating the formation of an anisotropic crystalline phase in the melt. The anisotropy of the crystalline phase, and consequently that of the final solid, depends on the deformation in the melt at the instant of crystallization, a fact that has been known for a long time and has been exploited in polymer processing. The proposed model is tested by solving three homogenous deformations.
Functional analysis
Probability theory and stochastic processes
73
94
10.4171/IFB/13
http://www.ems-ph.org/doi/10.4171/IFB/13
Simulation of dendritic crystal growth with thermal convection
Eberhard
Bänsch
Universität Erlangen-Nünberg, ERLANGEN, GERMANY
Alfred
Schmidt
Universität Freiburg, FREIBURG I BR, GERMANY
Dendritic solidification; crystal growth; natural convection; Gibbs-Thomson; Navier-Stokes equations; finite elements
The dendritic growth of crystals under gravity influence shows a strong dependence on convection in the liquid. The situation is modelled by the Stefan problem with a Gibbs-Thomson condition coupled with the Navier-Stokes equations in the liquid phase. A finite element method for the numerical simulation of dendritic crystal growth including convection effects is presented. It consists of a parametric finite element method for the evolution of the interface, coupled with finite element solvers for the heat equation and Navier-Stokes equations in a time dependent domain. Results from numerical simulations in two space dimensions with Dirichlet and transparent boundary conditions are included.
Functional analysis
Probability theory and stochastic processes
95
115
10.4171/IFB/14
http://www.ems-ph.org/doi/10.4171/IFB/14
2
Error bounds for a difference scheme approximating viscosity solutions of mean curvature flow
Klaus
Deckelnick
Otto-von-Guericke-Universität Magdeburg, MAGDEBURG, GERMANY
finite difference schemes; mean curvature flow
We analyse a finite difference scheme for the approximation of level set solutions to mean curvature flow. The scheme which was proposed by Crandall & Lions (Numer. Math. 75, (1996) 17-41) is a monotone and consistent discretization of a regularized version of the underlying problem. We derive an L[infin]-error bound between the numerical solution and the viscosity solution to the level set equation provided that the space and time step sizes are appropriately related to the regularization parameter.
Functional analysis
Probability theory and stochastic processes
117
142
10.4171/IFB/15
http://www.ems-ph.org/doi/10.4171/IFB/15
Flux pinning and boundary nucleation of vorticity in a mean field model of superconducting vortices
Charles
Elliott
University of Warwick, COVENTRY, UNITED KINGDOM
Vanessa
Styles
University of Sussex, BRIGHTON, UNITED KINGDOM
Flux pinning; boundary nucleation; mean field model; superconductivity
We study a one-dimensional mean field model of superconducting vortices with a finite London penetration depth, flux pinning and nucleation of vorticity at inflow boundary sections. The existence of a unique weak solution is proved and the long time behaviour is studied. A numerical discretization of the model is derived and it is shown that as the time step and the mesh size tend to zero, the discrete solution converges to the unique weak solution of the continuous model. Some numerical computations are presented which illustrate the effects of flux pinning and the finite penetration depth.
Functional analysis
Probability theory and stochastic processes
143
180
10.4171/IFB/16
http://www.ems-ph.org/doi/10.4171/IFB/16
Distribution of vortices in a type-II superconductor as a free boundary problem: existence and regularity via Nash-Moser theory
Alexis
Bonnet
Université de Cergy-Pontoise, CERGY-PONTOISE CEDEX, FRANCE
Régis
Monneau
Cité Descartes - Champs sur Marne, MARNE-LA-VALLÉE CEDEX 2, FRANCE
superconductor; free boundary problems
This paper is concerned with a model describing the distribution of vortices in a Type-II superconductor. These vortices are distributed continuously and occupy an unknown region D with [part]D representing the free boundary. The problem is set as follows: two constants H0 > H1 > 0 are given, to find an open subset D of the smooth bounded open set [ohm] [sub] R2 and a function H defined on [ohm]\D such that {div(F(|[nabla] H|2) [nabla] H) - H=0 in [ohm]\D where the function F is analytic positive increasing H=H0 on [par][ohm] H=H1 on [par]D [par]H[horbar][par]n = [par]D. Here we prove the existence of a solution with a domain D having an analytic boundary. We use the Nash-Moser inverse function theorem applied to a degenerate case.
Functional analysis
Probability theory and stochastic processes
181
200
10.4171/IFB/17
http://www.ems-ph.org/doi/10.4171/IFB/17
On a constrained variational problem with an arbitrary number of free boundaries
Paolo
Tilli
Politecnico di Torino, TORINO, ITALY
constrained variational problems; free boundary problems; immiscible fluids
We study the problem of minimizing the Dirichlet integral among all functions u [isin] H1([ohm]) whose level sets {u=li} have prescribed Lebesgue measure [agr]i. This problem was introduced in connection with a model for the interface between immiscible fluids. The existence of minimizers is proved with an arbitrary number of level-set constraints, and their regularity is investigated. Our technique consists in enlarging the class of admissible functions to the whole space H1([ohm]), penalizing those functions whose level sets have measures far from those required; in fact, we study the minimizers of a family of penalized functionals F[lgr], [lgr] > 0, showing that they are Höder continuous, and then we prove that such functions minimize the original functional also, provided the penalization parameter [lgr] is large enough. In the case where only two levels are involved, we prove Lipschitz continuity of the minimizers.
Functional analysis
Probability theory and stochastic processes
201
212
10.4171/IFB/18
http://www.ems-ph.org/doi/10.4171/IFB/18
3
On the asymptotic behaviour of anisotropic energies arising in the cardiac bidomain model
Luigi
Ambrosio
Scuola Normale Superiore, PISA, ITALY
Piero
Colli Franzone
Università di Pavia, PAVIA, ITALY
Giuseppe
Savaré
Università di Pavia, PAVIA, ITALY
localized minimization problems; anisotropic energies; cardiac bidomain model
We study the [Gamma]-convergence of a family of vectorial integral functionals, which are the sum of a vanishing anisotropic quadratic form in the gradients and a penalizing double-well potential depending only on a linear combination of the components of their argument. This particular feature arises from the study of the so-called `bidomain model' for the cardiac electric field; one of its consequences is that the L1-norm of a minimizing sequence can be unbounded and therefore a lack of coercivity occurs. We characterize the [Gamma]-limit as a surface integral functional, whose integrand is a convex function of the normal and can be computed by solving a localized minimization problem.
Functional analysis
Probability theory and stochastic processes
213
266
10.4171/IFB/19
http://www.ems-ph.org/doi/10.4171/IFB/19
Global existence for a non-local mean curvature flow as a limit of a parabolic-elliptic phase transition model
Danielle
Hilhorst
Université Paris-Sud, ORSAY CX, FRANCE
Elisabeth
Logak
Université de Cergy-Pontoise, CERGY-PONTOISE CEDEX, FRANCE
Reiner
Schätzle
Universität Tübingen, TÜBINGEN, GERMANY
free boundary problems; mean curvature; reaction-diffusion systems; microphase separation; diblock copolymers
We consider a free boundary problem where the velocity depends on the mean curvature and on some non-local term. This problem arises as the singular limit of a reaction-diffusion system which describes the microphase separation of diblock copolymers. The interface may present singularities in finite time. This leads us to consider weak solutions on an arbitrary time interval and to prove the global-in-time convergence of solutions of the reaction-diffusion system.
Functional analysis
Probability theory and stochastic processes
267
282
10.4171/IFB/20
http://www.ems-ph.org/doi/10.4171/IFB/20
Evolution of compressible and incompressible fluids separated by a closed interface
Irina
Denisova
Russian Acadademy of Sciences, ST. PETERSBURG, RUSSIAN FEDERATION
Free boundary problem, Navier-Stokes equations, two immiscible fluids
This work solves the problem governing the simultaneous motion of two viscous liquids of different kinds: compressible and incompressible. The boundary between the fluids is considered as an unknown (free) interface where the surface tension is taken into account. Although the fluids occupy the whole space 3, one of them should have a finite volume. Local (in time) unique solvability of this problem is obtained in the Sobolev-Slobodetski[inodot]spaces of functions. Estimates of the solution of a model problem for the Stokes equations are considered in detail, the interface between the fluids being a plane. The Schauder method is used to study a linear problem with a compact boundary. The passage to the nonlinear problem is made by successive approximations.
Functional analysis
Probability theory and stochastic processes
283
312
10.4171/IFB/21
http://www.ems-ph.org/doi/10.4171/IFB/21
Thermo-kinetically controlled pattern selection
Michael
Frankel
Indiana University Purdue University Indianapolis, INDIANAPOLIS, UNITED STATES
Laura Kathryn
Gross
University of Akron, AKRON, UNITED STATES
Victor
Roytburd
Rensselaer Polytechnic Institute, TROY, UNITED STATES
bifurcation and pattern formation; free boundary problems
Through a combination of asymptotic and numerical approaches we investigate bifurcation and pattern formation for a free boundary model related to a rapid crystallization of amorphous films and to the self-propagating high-temperature synthesis (solid combustion). The unifying feature of these diverse physical phenomena is the existence of a uniformly propagating wave of phase transition whose stability is controlled by the balance between the energy production at the interface and the energy dissipation into the medium. For the propagation on a two-dimensional strip with thermally insulated edges, we develop a multi-scale weakly-nonlinear analysis that results in a system of ordinary differential equations for the slowly varying amplitudes. We identify a nonlinear parameter which is responsible for the pattern selection, and utilize the amplitude system for predicting the evolving patterns. The pattern selection is confirmed by direct numerical simulations on the free boundary problem. Some numerical results on strongly nonlinear regimes are also presented.
Functional analysis
Probability theory and stochastic processes
313
330
10.4171/IFB/22
http://www.ems-ph.org/doi/10.4171/IFB/22
On Maxwellian equilibria of insulated semiconductors
Luis
Caffarelli
University of Texas at Austin, AUSTIN, UNITED STATES
Jean
Dolbeault
Université de Paris Dauphine, PARIS CEDEX 16, FRANCE
Peter
Markowich
Universität Wien, WIEN, AUSTRIA
Christian
Schmeiser
TU Wien, WIEN, AUSTRIA
Semiconductors, equilibrium, free boundary problems, charge neutrality
A semi-linear elliptic integro-differential equation subject to homogeneous Neumann boundary conditions for the equilibrium potential in an insulated semiconductor device is considered. A variational formulation gives existence and uniqueness. The limit as the scaled Debye length tends to zero is analysed. Two different cases occur. If the number of free electrons and holes is sufficiently high, local charge neutrality prevails throughout the device. Otherwise, depletion regions occur, and the limiting potential is the solution of a free boundary problem.
Functional analysis
Probability theory and stochastic processes
331
339
10.4171/IFB/23
http://www.ems-ph.org/doi/10.4171/IFB/23
4
Error estimates for a semi-implicit fully discrete finite element scheme for the mean curvature flow of graphs
Klaus
Deckelnick
Otto-von-Guericke-Universität Magdeburg, MAGDEBURG, GERMANY
Gerhard
Dziuk
Universität Freiburg, FREIBURG I BR, GERMANY
finite element scheme, mean curvature flow
The efficient numerical simulation of the curvature-driven motion of interfaces is an important tool in several free- boundary problems. We treat the case of an interface which is given as a graph. The highly non-linear problem is discretized in space by piecewise linear finite elements. Although the problem is not in divergence form it can be written in a variational form which allows the use of the modern adaptive techniques of finite elements. The time discretization is carried out in a semi-implicit way such that in every time step a linear system with symmetric positive matrix has to be solved. Optimal error estimates are proved for the fully discrete problem under the assumption that the time-step size is bounded by the spatial grid size.
Functional analysis
Probability theory and stochastic processes
341
359
10.4171/IFB/24
http://www.ems-ph.org/doi/10.4171/IFB/24
On convergence of solutions of the crystalline Stefan problem with Gibbs-Thomson law and kinetic undercooling
Piotr
Rybka
Warsaw University, WARSZAWA, POLAND
Free boundary, Stefan problem, Gibbs-Thomson law, crystalline anisotropy
This paper presents a study of the relations between the modified Stefan problem in a plane and its quasi-steady approximation. In both cases the interfacial curve is assumed to be a polygon. It is shown that the weak solutions to the Stefan problem converge to weak solutions of the quasi-steady problem as the bulk specific heat tends to zero. The initial interface has to be convex of sufficiently small perimeter.
Functional analysis
Probability theory and stochastic processes
361
379
10.4171/IFB/25
http://www.ems-ph.org/doi/10.4171/IFB/25
A free-boundary problem in combustion theory
Julián
Fernández Bonder
Universidad de Buenos Aires, BUENOS AIRES, ARGENTINA
Noemi
Wolanski
Universidad de Buenos Aires, BUENOS AIRES, ARGENTINA
Free boundary; Stefan problem; Gibbs-Thomson law; crystalline anisotropy
This paper presents a study of the relations between the modified Stefan problem in a plane and its quasi-steady approximation. In both cases the interfacial curve is assumed to be a polygon. It is shown that the weak solutions to the Stefan problem converge to weak solutions of the quasi-steady problem as the bulk specific heat tends to zero. The initial interface has to be convex of sufficiently small perimeter.
Functional analysis
Probability theory and stochastic processes
381
411
10.4171/IFB/26
http://www.ems-ph.org/doi/10.4171/IFB/26
A free-boundary problem for Stokes equations: classical solutions
Stanislav
Antontsev
Universidade de Lisboa, LISBOA, PORTUGAL
Anvarbek
Meirmanov
Belgorod State University, BELGOROD, RUSSIAN FEDERATION
Vadim
Yurinsky
Universidade da Beira Interior, COVILHÃ, PORTUGAL
free-boundary problem, immiscible viscous fluids
The problem considered is that of evolution of the free boundary separating two immiscible viscous fluids with different constant densities. The motion is described by the Stokes equations driven by the gravity force. For flows in a bounded domain &OHgr; ?n, n ? 2, we prove existence and uniqueness of classical solutions, and concentrate on the study of properties of the moving boundary separating the two fluids.
Functional analysis
Probability theory and stochastic processes
413
424
10.4171/IFB/27
http://www.ems-ph.org/doi/10.4171/IFB/27
Behaviour of interfaces in a diffusion-absorption equation with critical exponents
Victor
Galaktionov
University of Bath, BATH, UNITED KINGDOM
Sergey
Shmarev
Universidad de Oviedo, OVIEDO, SPAIN
Juan Luis
Vázquez
Universidad Autónoma de Madrid, MADRID, SPAIN
Non-linear diffusion equation; strong absorption; interface equation; regularity; analyticity; turning points; matching expansion
We consider the Cauchy problem for the porous medium equation with strong absorption ut= (um)xx? upforx ? R, t & 0, with continuous compactly supported initial data u(x, 0) = u0(x) ? 0in the critical case m + p = 2of the range of parameters m & 1, p < 1. We study the regularity of solutions and interfaces and compare the results with the purely diffusive case ut= (um)xx. Important differences are found in the interface behaviour and equations, in the occurrence of turning points and inflection points of the interfaces, and in the fact that bounded solutions extinguish in finite time. All these phenomena are examined and described. The critical case studied here allows for a comparatively simpler and richer analysis of the qualitative behaviour of non-linear parabolic equations involving finite propagation (m & 1) and strong absorption (p < 1).
Functional analysis
Probability theory and stochastic processes
425
448
10.4171/IFB/28
http://www.ems-ph.org/doi/10.4171/IFB/28
The `hump' effect in solid propellant combustion
Gawtum
Namah
Université de Franche-Comté, BESANCON CEDEX, FRANCE
Jean-Michel
Roquejoffre
Université Paul Sabatier, TOULOUSE CEDEX, FRANCE
Level set formulation, travelling fronts, homogenization, time-asymptotic stabilization
An eikonal equation modelling the propagation of combustion fronts in striated media is studied via a level set formulation. Travelling fronts are obtained, and their speeds turn out to be monotone with respect to the angle of the striations. An effective equation for thin meniscus striations is derived from a homogenization process, thus explaining the so-called `hump' effect. Finally, the time-asymptotic stabilization of unsteady fronts propagating in straight striations is proved.
Functional analysis
Probability theory and stochastic processes
449
467
10.4171/IFB/29
http://www.ems-ph.org/doi/10.4171/IFB/29