- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 07:36:00
22
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=12&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
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
12
2010
1
Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves
Pietro
Baldi
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
John
Toland
University of Bath, BATH, UNITED KINGDOM
Hydrodynamic waves, hydroelastic waves, nonlinear elasticity, free boundary problems, travelling waves, bifurcation theory, secondary bifurcations, Wilton ripples, Lyapunov–Schmidt reduction, symmetry-breaking
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
1
22
10.4171/IFB/224
http://www.ems-ph.org/doi/10.4171/IFB/224
Transport of interfaces with surface tension by 2D viscous flows
David
Ambrose
Drexel University, PHILADELPHIA, UNITED STATES
Milton
Lopes Filho
IMECC - UNICAMP, CAMPINAS, BRAZIL
Helena
Nussenzveig Lopes
IMECC - UNICAMP, CAMPINAS, BRAZIL
Walter
Strauss
Brown University, PROVIDENCE, UNITED STATES
We consider the problem of finding a global weak solution for two-dimensional, incompressible viscous flow on a torus, containing a surface-tension bearing curve transported by the flow. This is the simplest case of a class of two-phase flows considered by Plotnikov in [16] and Abels in [1]. Our work complements Abels’ analysis by examining this special case in detail. We construct a family of approximations and show that the limit of these approximations satisfies, globally in time, an incomplete set of equations in the weak sense. In addition, we examine criteria for closure of the limit system, we find conditions which imply nontrivial dependence of the limiting solution on the surface tension parameter, and we obtain a new system of evolution equations which models our flow-interface problem, in a form that may be useful for further analysis and for numerical simulations.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
23
44
10.4171/IFB/225
http://www.ems-ph.org/doi/10.4171/IFB/225
A posteriori error controlled local resolution of evolving interfaces for generalized Cahn–Hilliard equations
Sören
Bartels
Universität Bonn, BONN, GERMANY
Rüdiger
Müller
Angewandte Analysis und Stochastik, BERLIN, GERMANY
For equations of generalized Cahn–Hilliard type we present an a posteriori error analysis that is robust with respect to a small interface length scale γ. We propose the solution of a fourth order elliptic eigenvalue problem in each time step to gain a fully computable error bound, which only depends polynomially (of low order) on the inverse of γ. A posteriori and a priori error bounds for the eigenvalue problem are also derived. In numerical examples we demonstrate that this approach extends the applicability of robust a posteriori error estimation as it removes restrictive conditions on the initial data. Moreover we show that the computation of the principal eigenvalue allows the detection of critical points during the time evolution that limit the validity of the estimate.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
45
74
10.4171/IFB/226
http://www.ems-ph.org/doi/10.4171/IFB/226
Convexity breaking of the free boundary for porous medium equations
Kazuhiro
Ishige
Tohoku University, SENDAI, JAPAN
Paolo
Salani
Università degli Studi di Firenze, FIRENZE, ITALY
We investigate the preservation of convexity of the free boundary by the solutions of the porous medium equation. We prove that starting with an initial datum with some kind of suboptimal αconcavity property, the convexity of the positivity set can be lost in a short time.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
75
84
10.4171/IFB/227
http://www.ems-ph.org/doi/10.4171/IFB/227
Mixed finite element method for electrowetting on dielectric with contact line pinning
Shawn
Walker
Louisiana State University, BATON ROUGE, UNITED STATES
Andrea
Bonito
Texas A&M University, COLLEGE STATION, UNITED STATES
Ricardo
Nochetto
University of Maryland, COLLEGE PARK, UNITED STATES
We present a mixed finite element method for a model of the flow in a Hele-Shaw cell of 2-D fluid droplets surrounded by air driven by surface tension and actuated by an electric field. The application of interest regards a micro-fluidic device called ElectroWetting on Dielectric (EWOD). Our analysis first focuses on the time discrete (continuous in space) problem and is presented in a mixed variational framework, which incorporates curvature as a natural boundary condition. The model includes a viscous damping term for interface motion, as well as contact line pinning (sticking of the interface) and is captured in our formulation by a variational inequality. The semi-discrete problem uses a semiimplicit time discretization of curvature. We prove the well-posedness of the semi-discrete problem and fully discrete problem when discretized with iso-parametric finite elements. We derive a priori error estimates for the space discretization. We also prove the convergence of an Uzawa algorithm for solving the semi-discrete EWOD system with inequality constraint. We conclude with a discussion about experimental orders of convergence.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
85
119
10.4171/IFB/228
http://www.ems-ph.org/doi/10.4171/IFB/228
2
Well-posedness of a parabolic moving-boundary problem in the setting of Wasserstein gradient flows
Jacobus
Portegies
New York University, NEW YORK, UNITED STATES
Mark
Peletier
Eindhoven University of Technology, EINDHOVEN, NETHERLANDS
We develop a gradient-flow framework based on the Wasserstein metric for a parabolic moving boundary problem that models crystal dissolution and precipitation. In doing so we derive a new weak formulation for this moving-boundary problem and we show that this formulation is well posed. In addition, we develop a new uniqueness technique based on the framework of gradient flows with respect to the Wasserstein metric. With this uniqueness technique, the Wasserstein framework becomes a complete well-posedness setting for this parabolic moving-boundary problem.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
121
150
10.4171/IFB/229
http://www.ems-ph.org/doi/10.4171/IFB/229
Stable constant-mean-curvature hypersurfaces are area minimizing in small L1 neighborhoods
Frank
Morgan
Williams College, WILLIAMSTOWN, UNITED STATES
Antonio
Ros
Universidad de Granada, GRANADA, SPAIN
We prove that a strictly stable oriented constant-mean-curvature hypersurface in a smooth closed manifold of dimension less than or equal to 7 is uniquely homologically area minimizing for fixed volume in a small L1 neighborhood, proving a conjecture of Choksi and Sternberg.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
151
155
10.4171/IFB/230
http://www.ems-ph.org/doi/10.4171/IFB/230
On a free boundary problem describing the phase transition in an incompressible viscous fluid
Yoshiaki
Kusaka
Tamagawa University, TOKYO, JAPAN
Ice melts at 0◦ C under a pressure of 1 atm, and increasing the pressure decreases the melting temperature. In the present paper, a new problem is posed that describes the process of phase transition in an incompressible viscous fluid, taking into account the above-described pressure effect. This problem is described as a free boundary problem in terms of the Navier–Stokes equations coupled with the heat equation, where the equilibrium temperature is assumed to be related to the pressure by the Clapeyron–Clausius equation. We prove the existence of a global-in-time solution.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
157
185
10.4171/IFB/231
http://www.ems-ph.org/doi/10.4171/IFB/231
Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies
John
Barrett
Imperial College London, LONDON, UNITED KINGDOM
Harald
Garcke
Universität Regensburg, REGENSBURG, GERMANY
Robert
Nürnberg
Imperial College London, LONDON, UNITED KINGDOM
Surface cluster, mean curvature flow, surface diffusion, soap bubbles, triple junction lines, parametric finite elements, anisotropy, tangential movement
We present a variational formulation for the evolution of surface clusters in ℝ3 by mean curvature flow, surface diffusion and their anisotropic variants. We introduce the triple junction line conditions that are induced by the considered gradient flows, and present weak formulations of these flows. In addition, we consider the case where a subset of the boundaries of these clusters are constrained to lie on an external boundary. These formulations lead to unconditionally stable, fully discrete, parametric finite element approximations. The resulting schemes have very good properties with respect to the distribution of mesh points and, if applicable, volume conservation. This is demonstrated by several numerical experiments, including isotropic double, triple and quadruple bubbles, as well as clusters evolving under anisotropic mean curvature flow and anisotropic surface diffusion, including computations for regularized crystalline surface energy densities.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
187
234
10.4171/IFB/232
http://www.ems-ph.org/doi/10.4171/IFB/232
A free boundary problem arising in a simplified tumour growth model of contact inhibition
Michiel
Bertsch
Consiglio Nazionale delle Ricerche, ROMA, ITALY
Roberta
Dal Passo
Università di Roma, ROMA, ITALY
Masayasu
Mimura
Meiji University, KAWASAKI, JAPAN
It is observed in vitro and in vivo that when two populations of different types of cells come near to each other, the rate of proliferation of most cells decreases. This phenomenon is often called contact inhibition of growth between two cells. In this paper, we consider a simplified 1-dimensional PDE model for normal and abnormal cells, motivated by the paper by Chaplain, Graziano and Preziosi ([5]). We show that if the two populations are initially segregated, then they remain segregated due to the contact inhibition mechanism. In this case the system of PDE’s can be formulated as a free boundary problem.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
235
250
10.4171/IFB/233
http://www.ems-ph.org/doi/10.4171/IFB/233
On the existence of mean curvature flow with transport term
Chun
Liu
The Pennsylvania State University, UNIVERSITY PARK, UNITED STATES
Norifumi
Sato
Furano H.S., FURANO (HOKKAIDO), JAPAN
Yoshihiro
Tonegawa
Hokkaido University, SAPPORO, JAPAN
Mean curvature flow, varifold, Allen–Cahn equation, phase field method
We prove the global-in-time existence of weak solution for a hypersurface evolution problem where the velocity is the sum of the mean curvature and arbitrarily given non-smooth vector field in a suitable Sobolev space. The approximate solution is obtained by the Allen–Cahn equation with transport term. By establishing the density ratio upper bound on the phase boundary measure it is shown that the limiting surface moves with the desired velocity in the sense of Brakke.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
251
277
10.4171/IFB/234
http://www.ems-ph.org/doi/10.4171/IFB/234
3
Boundary regularity for a parabolic obstacle type problem
John
Andersson
University of Warwick, COVENTRY, UNITED KINGDOM
We study the regularity of the free boundary, near contact points with the fixed boundary, for a parabolic free boundary problem Δu –∂u/∂t = χ{u≠0} in Qr+ = {(x, t) ∈ Br x (–r2, 0); x1 > 0}, u = f(x, t) on {x1 = 0} ∩ Qr. We will show that under certain regularity assumptions on the boundary data f the free boundary is a C1 manifold up to the fixed boundary. We also show that the C1 modulus of continuity is uniform for a certain, and specified, subclass of solutions.
Fluid mechanics
General
279
291
10.4171/IFB/235
http://www.ems-ph.org/doi/10.4171/IFB/235
Nonlinear stability analysis of a two-dimensional diffusive free boundary problem
Micah
Webster
Goucher College, BALTIMORE, UNITED STATES
Patrick
Guidotti
University of California, Irvine, IRVINE, UNITED STATES
We explore global existence and stability of planar solutions to a multi-dimensional Case II polymer diffusion model which takes the form of a one-phase free boundary problem with phase onset. Due to a particular boundary condition, convergence cannot be expected on the whole domain. A boundary integral formulation derived in [13] is shown to remain valid in the present context and allows us to circumvent this difficulty by restricting the analysis to the free boundary. The integral operators arising in the boundary integral formulation are analyzed by methods of pseudodifferential calculus. This is possible as explicit symbols are available for the relevant kernels. Spectral analysis of the linearization can then be combined with a known principle of linearized stability [12] to obtain local exponential stability of planar solutions with respect to two-dimensional perturbations.
General
293
310
10.4171/IFB/236
http://www.ems-ph.org/doi/10.4171/IFB/236
On the two-phase Navier–Stokes equations with surface tension
Gottfried
Anger
Martin-Luther-Universität Halle-Wittenberg, HALLE, GERMANY
Gieri
Simonett
Vanderbilt University, NASHVILLE, UNITED STATES
Navier–Stokes equations; surface tension; well-posedness; analyticity
The two-phase free boundary problem for the Navier–Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of Lp-maximal regularity of the underlying linear problem we show local well-posedness of the problem, and prove that the solution, in particular the interface, becomes instantaneously real analytic.
Partial differential equations
Fluid mechanics
General
311
345
10.4171/IFB/237
http://www.ems-ph.org/doi/10.4171/IFB/237
Cauchy problems for noncoercive Hamilton–Jacobi–Isaacs equations with discontinuous coefficients
Cecilia
De Zan
Università di Padova, PADOVA, ITALY
Pierpaolo
Soravia
Università di Padova, PADOVA, ITALY
We study the Cauchy problem for a homogeneous and not necessarily coercive Hamilton–Jacobi–Isaacs equation with an x-dependent, piecewise continuous coefficient. We prove that under suitable assumptions there exists a unique and continuous viscosity solution. The result applies in particular to the Carnot–Carathéodory eikonal equation with discontinuous refraction index of a family of vector fields satisfying the Hörmander condition. Our results are also of interest in connection with geometric flows with discontinuous velocity in anisotropic media with a non-euclidian ambient space.
Partial differential equations
Calculus of variations and optimal control; optimization
General
347
368
10.4171/IFB/238
http://www.ems-ph.org/doi/10.4171/IFB/238
Long-time behaviour of two-phase solutions to a class of forward-backward parabolic equations
Flavia
Smarazzo
Università di Roma "La Sapienza", ROMA, ITALY
Forward-backward equations; two-phase solutions; pseudoparabolic regularization; longtime behaviour of solutions; steady states.
We consider two-phase solutions to the Neumann initial-boundary value problem for the parabolic equation ut=[ϕ(u)]xx, where ϕ is a nonmonotone cubic-like function. First, we prove global existence for a restricted class of initial data u0, showing that two-phase solutions can be obtained as limiting points of the family of solutions to the Neumann initial-boundary value problem for the regularized equation utε = [ϕ(uε)]xx + εutxxε (ε > 0) . Then, assuming global existence, we study the long-time behaviour of two-phase solutions for any initial datum u0.
Partial differential equations
Measure and integration
General
369
408
10.4171/IFB/239
http://www.ems-ph.org/doi/10.4171/IFB/239
4
Convergence of a large time-step scheme for mean curvature motion
Elisabetta
Carlini
Università di Roma La Sapienza, ROMA, ITALY
Maurizio
Falcone
Università di Roma La Sapienza, ROMA, ITALY
R.
Ferretti
Università di Roma Tre, ROMA, ITALY
Mean curvature motion; level-set approach; semi-Lagrangian schemes; consistency; generalized monotonicity; convergence
We analyse the properties of a semi-Lagrangian scheme for the approximation of the Mean Curvature Motion (MCM). This approximation is obtained by coupling a stochastic method for the approximation of characteristics (to be understood in a generalized sense) with a local interpolation. The main features of the scheme are that it can handle degeneracies, it is explicit and it allows for large time steps. We also propose a modified version of this scheme, for which monotonicity and consistency can be proved. Then convergence to the viscosity solution of the MCM equation follows by an extension of the Barles–Souganidis theorem. The scheme is also compared with similar existing schemes proposed by Crandall and Lions and, more recently, by Kohn and Serfaty. Finally, several numerical test problems in 2D and 3D are presented.
Numerical analysis
Calculus of variations and optimal control; optimization
General
409
441
10.4171/IFB/240
http://www.ems-ph.org/doi/10.4171/IFB/240
The Neumann problem in an irregular domain
Łukasz
Bolikowski
University of Warsaw, WARSAW, POLAND
Maria
Gokieli
University of Warsaw, WARSAW, POLAND
Nicolas
Varchon
University of Warsaw, WARSAW, POLAND
We consider ‘patterns’ stability for the reaction-diffusion equation with Neumann boundary conditions in an irregular domain in ℝN, N ≥ 2, the model example being two convex regions connected by a small ‘hole’ in their boundaries. By patterns we mean solutions having an interface, i.e. a transition layer between two constants. It is well known that in 1D domains and in many 2D domains, patterns are unstable for this equation. We show that, unlike the 1D case, but as in 2D dumbbell domains, stable patterns exist. In a more general way, we prove invariance of stability properties for steady states when a sequence of domains Ωn converges to our limit domain in the sense of Mosco. We illustrate the theoretical results by numerical simulations of evolving and persisting interfaces.
General
443
462
10.4171/IFB/241
http://www.ems-ph.org/doi/10.4171/IFB/241
The thin film equation with backwards second order diffusion
Amy
Novick-Cohen
Technion - Israel Institute of Technology, HAIFA, ISRAEL
Andrey
Shishkov
Academy of Sciences of Ukraine, DONETSK, UKRAINE
Thin film equation; backwards diffusion; higher order parabolic equations; degenerate parabolic equations; finite speed of propagation
We focus on the thin film equation with lower order “backwards” diffusion which can describe, for example, the evolution of thin viscous films in the presence of gravity and thermo-capillary effects, or the thin film equation with a “porous media cutoff” of van der Waals forces. We treat in detail the equation ut + {un(uxxx+νu m–nux–AuM–nux)}x = 0, where ν = +/– 1, n > 0, M > m, and A ≥ 0. Global existence of weak nonnegative solutions is proven when m–n > –2 and A > 0 or ν = –1, and when –2 < m–n < 2, A = 0, ν = 1. From the weak solutions, we get strong entropy solutions under the additional constraint that m–n > –3/2 if ν = 1. A local energy estimate is obtained when 2 ≤ n < 3 under some additional restrictions. Finite speed of propagation is proven when m > n/2, for the case of “strong slippage”, 0 < n < 2, when ν = 1 based on local entropy estimates, and for the case of “weak slippage”, 2 ≤ n < 3, when ν = +/– 1 based on local entropy and energy estimates.
Partial differential equations
Fluid mechanics
General
463
496
10.4171/IFB/242
http://www.ems-ph.org/doi/10.4171/IFB/242
Wavelet analogue of the Ginzburg–Landau energy and its Γ-convergence
Julia
Dobrosotskaya
University of Maryland, COLLEGE PARK, UNITED STATES
Andrea
Bertozzi
University of California Los Angeles, LOS ANGELES, UNITED STATES
This paper considers a wavelet analogue of the classical Ginzburg–Landau energy, where the H1 seminorm is replaced by the Besov seminorm defined via an arbitrary regular wavelet. We prove that functionals of this type Γ-converge to a weighted analogue of the TV functional on characteristic functions of finite-perimeter sets. The Γ-limiting functional is defined explicitly, in terms of the wavelet that is used to define the energy. We show that the limiting energy is none other than the surface tension energy in the 2D Wulff problem and its minimizers are represented by the corresponding Wulff shapes. This fact as well as the
General
497
525
10.4171/IFB/243
http://www.ems-ph.org/doi/10.4171/IFB/243
Mass conserving Allen–Cahn equation and volume preserving mean curvature flow
Xinfu
Chen
University of Pittsburgh, PITTSBURGH, UNITED STATES
Danielle
Hilhorst
Université Paris-Sud, ORSAY CX, FRANCE
Elisabeth
Logak
Université de Cergy-Pontoise, CERGY-PONTOISE CEDEX, FRANCE
We consider a mass conserving Allen–Cahn equation ut = Δ u + ε–2(f(u) – ελ(t)) in a bounded domain with no flux boundary condition, where ελ(t) is the average of f(u(∙,t)) and –f is the derivative of a double equal well potential. Given a smooth hypersurface ⓾ contained in the domain, we show that the solution uε with appropriate initial data tends, as ε ↘ 0, to a limit which takes only two values, with the jump occurring at the hypersurface obtained from the volume preserving mean curvature flow starting from ⓾.
General
527
549
10.4171/IFB/244
http://www.ems-ph.org/doi/10.4171/IFB/244
Error analysis for the approximation of axisymmetric Willmore flow by C1-finite elements
Klaus
Deckelnick
Otto-von-Guericke-Universität Magdeburg, MAGDEBURG, GERMANY
Friedhelm
Schieweck
Otto-von-Guericke-Universität Magdeburg, MAGDEBURG, GERMANY
Willmore flow; Dirichlet boundary conditions; finite elements; error estimates
We consider the Willmore flow of axially symmetric surfaces subject to Dirichlet boundary conditions. The corresponding evolution is described by a nonlinear parabolic PDE of fourth order for the radius function. A suitable weak form of the equation, which is based on the first variation of the Willmore energy, leads to a semidiscrete scheme, in which we employ piecewise cubic C1-finite elements for the one-dimensional approximation in space.We prove optimal error bounds in Sobolev norms for the solution and its time derivative and present numerical test examples.
Partial differential equations
Numerical analysis
General
551
574
10.4171/IFB/245
http://www.ems-ph.org/doi/10.4171/IFB/245