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  1. Nonlinear Differential Equations and Applications NoDEA
  2. Nonlinear Differential Equations and Applications NoDEA : Volume 20
  3. Nonlinear Differential Equations and Applications NoDEA : Volume 20, Issue 3, June 2013
  4. Passing from bulk to bulk-surface evolution in the Allen–Cahn equation
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Nonlinear Differential Equations and Applications NoDEA : Volume 23
Nonlinear Differential Equations and Applications NoDEA : Volume 22
Nonlinear Differential Equations and Applications NoDEA : Volume 21
Nonlinear Differential Equations and Applications NoDEA : Volume 20
Nonlinear Differential Equations and Applications NoDEA : Volume 20, Issue 6, December 2013
Nonlinear Differential Equations and Applications NoDEA : Volume 20, Issue 5, October 2013
Nonlinear Differential Equations and Applications NoDEA : Volume 20, Issue 4, August 2013
Nonlinear Differential Equations and Applications NoDEA : Volume 20, Issue 3, June 2013
Lax type formulas with lower semicontinuous initial data and hypercontractivity results
Hamilton–Jacobi equations constrained on networks
The strong relaxation limit of the multidimensional Euler equations
Local Hölder continuity of weak solutions for an anisotropic elliptic equation
Erratum to: Local Hölder continuity of weak solutions for an anisotropic elliptic equation
Positive solution to the nonlinear abstract Hammerstein equation and applications to $${\phi}$$ -Laplacian BVPs
Nonlinear Schrödinger problems: symmetries of some variational solutions
Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow
Existence and stability of solutions for a strongly coupled system modelling thin fluid films
Volterra equations in Banach spaces with completely monotone kernels
One multidimensional version of the Darboux first problem for one class of semilinear second order hyperbolic systems
Mean curvature flow with a constant forcing
The first Darboux problem for wave equations with nonlinear dissipative term
Effect of hyperviscosity on the geometry of the vorticity
A weighted eigencurve for Steklov problems with a potential
Existence and symmetry results for competing variational systems
Non-uniform dependence on initial data for the periodic modified Camassa–Holm equation
Strong solutions of semilinear stochastic partial differential equations
On strictly positive solutions for some semilinear elliptic problems
Existence of weak solutions for the generalized Navier–Stokes equations with damping
Planar Hamiltonian systems at resonance: the Ahmad–Lazer–Paul condition
Singular dynamics under a weak potential on a sphere
Self-similar solutions for a class of non-divergence form equations
(L ∞ +  Bolza) control problems as dynamic differential games
Passing from bulk to bulk-surface evolution in the Allen–Cahn equation
Regular and irregular solutions for a class of elliptic systems in the critical dimension
Existence of entire solutions for a class of quasilinear elliptic equations
On the stochastic 2-D motion of a Bingham fluid
Existence and multiplicity for positive solutions of a system of higher-order multi-point boundary value problems
Unconditional uniqueness in the charge class for the Dirac–Klein–Gordon equations in two space dimensions
Mathematical modeling of rigid bodies motion through some non-Newtonian flows
Hölder estimates for trajectories of differential inclusions and HJB equations with state constraints
Positive and non-positive solutions for an inviscid dyadic model: well-posedness and regularity
Quasilinear elliptic and parabolic Robin problems on Lipschitz domains
The Cauchy problem for the Novikov equation
Convergence of bounded solutions of nonlinear parabolic problems on a bounded interval: the singular case
On the supercritical KdV equation with time-oscillating nonlinearity
Long time behavior of solutions to semilinear parabolic equations involving strongly degenerate elliptic differential operators
Optimality conditions for reflecting boundary control problems
Nonlinearly coupled thermo-visco-elasticity
Rigorous justification of the short-pulse equation
Liouville theorems for fully nonlinear elliptic equations on spherically symmetric Riemannian manifolds
Saddle solutions for bistable symmetric semilinear elliptic equations
On the existence of periodic and homoclinic orbits for first order superquadratic Hamiltonian systems
Non-resonant boundary value problems with singular $${\phi}$$ -Laplacian operators
Critical exponent for semilinear wave equation with critical potential
Large time behavior for the heat equation on Carnot groups
Erratum to: A pointwise constrained version of the Liapunov convexity theorem for single integrals
Nonlinear Differential Equations and Applications NoDEA : Volume 20, Issue 2, April 2013
Nonlinear Differential Equations and Applications NoDEA : Volume 20, Issue 1, February 2013
Nonlinear Differential Equations and Applications NoDEA : Volume 19
Nonlinear Differential Equations and Applications NoDEA : Volume 18
Nonlinear Differential Equations and Applications NoDEA : Volume 17
Nonlinear Differential Equations and Applications NoDEA : Volume 16
Nonlinear Differential Equations and Applications NoDEA : Volume 15
Nonlinear Differential Equations and Applications NoDEA : Volume 14
Nonlinear Differential Equations and Applications NoDEA : Volume 13
Nonlinear Differential Equations and Applications NoDEA : Volume 12
Nonlinear Differential Equations and Applications NoDEA : Volume 11
Nonlinear Differential Equations and Applications NoDEA : Volume 10
Nonlinear Differential Equations and Applications NoDEA : Volume 9
Nonlinear Differential Equations and Applications NoDEA : Volume 8
Nonlinear Differential Equations and Applications NoDEA : Volume 7
Nonlinear Differential Equations and Applications NoDEA : Volume 6
Nonlinear Differential Equations and Applications NoDEA : Volume 5
Nonlinear Differential Equations and Applications NoDEA : Volume 4

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Passing from bulk to bulk-surface evolution in the Allen–Cahn equation

Content Provider SpringerLink
Author Liero, Matthias
Copyright Year 2012
Abstract In this paper we formulate a boundary layer approximation for an Allen–Cahn-type equation involving a small parameter $${\varepsilon}$$ . Here, $${\varepsilon}$$ is related to the thickness of the boundary layer and we are interested in the limit $${\varepsilon \to 0}$$ in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L2-gradient flow with the corresponding Allen–Cahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Γ- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.
Ending Page 942
Page Count 24
Starting Page 919
File Format PDF
ISSN 10219722
e-ISSN 14209004
Journal Nonlinear Differential Equations and Applications NoDEA
Issue Number 3
Volume Number 20
Language English
Publisher SP Birkhäuser Verlag Basel
Publisher Date 2012-08-07
Publisher Place Basel
Access Restriction One Nation One Subscription (ONOS)
Subject Keyword Energy balance Initial-boundary value problems for second-order parabolic equations Analysis Γ-convergence Gradient flow Mosco convergence Dynamic boundary condition Dynamic and nonequilibrium phase transitions (general) Allen–Cahn equation Nonlinear parabolic equations
Content Type Text
Resource Type Article
Subject Applied Mathematics Analysis
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