boundary effects on localized structures in spatially extended systems abstract we present a general method of analyzing the influence of finite size and boundary effects on the dynamics of localized solutions of non linear spatially extended systems the dynamics of localized structures in infinite systems involve solvability conditions that require projection onto a goldstone mode our method works by extending the solvability conditions to finite sized systems by incorporating the finite sized modifications of the goldstone mode and associated nonzero eigenvalue we apply this method to the special case of non equilibrium domain walls under the influence of dirichlet boundary conditions in a parametrically forced complex ginzburg landau equation where we examine exotic nonuniform domain wall motion due to the influence of boundary conditions

10 1007 s10955 006 9040 z we derive hydrodynamic equations describing the evolution of a binary fluid segregated into two regions each rich in one species which are separated on the macroscopic scale by a sharp interface our starting point is a vlasov boltzmann vb equation describing the evolution of the one particle position and velocity distributions fi x v t i 1 2 the solution of the vb equation is developed in a hilbert expansion appropriate for this system this yields incompressible navier stokes equations for the velocity field u and a jump boundary condition for the pressure across the interface the interface in turn moves with a velocity given by the normal component of u

canonical variables and analysis on so n 2 the approach of berezin to the quantization of so n 2 via generalized coherent states is considered in detail a family of n commuting observables is found in which the basis for an associated fock type representation space is expressed an interesting feature is that computations can be done by explicit matrix calculations in a particular basis the basic technical tool is the leibniz function the inner product of coherent states

front motion for phase transitions in systems with memory abstract we consider the following partial integro differential equation allen cahn equation with memory 2 t 0 t a t t 2 f h t d t where is a small parameter h a constant f the negative derivative of a double well potential and the kernel a is a piecewise continuous differentiable at the origin scalar valued function on 0 the prototype kernels are exponentially decreasing functions of time and they reduce the integro differential equation to a hyperbolic one the damped klein gordon equation by means of a formal asymptotic analysis we show that to the leading order and under suitable assumptions on the kernels the integro differential equation behaves like a hyperbolic partial differential equation obtained by considering prototype kernels the evolution of fronts is governed by the extended damped born infeld equation we also apply our method to a system of partial integro differential equations which generalize the classical phase field equations with a non conserved order parameter and describe the process of phase transitions where memory effects are present u t 2 t 0 t a 1 t t u t d t 2 t 0 t a 2 t t 2 f u t d t where is a small parameter in this case the functions u and represent the temperature field and order parameter respectively the kernels a1 and a2 are assumed to be similar to a for the phase field equations with memory we obtain the same result as for the generalized klein gordon equation or allen cahn equation with memory

motion by curvature in generalized cahn allen models the cahn allen model for the motion of phase antiphase boundaries is generalized to account for nonlinearities in the kinetic coefficient relaxation velocity and the coefficient of the gradient free energy the resulting equation is varepsilon 2 u l alpha u varepsilon kappa u raise0 5ex hbox scriptstyle 1 kern 0 1em kern 0 15em lower0 25ex hbox scriptstyle 2 nabla cdot kappa u raise0 5ex hbox scriptstyle 1 kern 0 1em kern 0 15em lower0 25ex hbox scriptstyle 2 nabla u f u wheref is bistable hereu is an order parameter and and are physical quantities associated with the system s free energy and relaxation speed respectively grain boundaries away from triple junctions are modeled by solutions with internal layers when e 1 the classical motion by curvature law for solution layers well known when and are constant is shown by formal asymptotic analysis to be unchanged in form under this generalization the only difference being in the value of the coefficient entering into the relation the analysis is extended to the case when the relaxation time for the process vanishes for a set of values ofu then is infinite for those values

crystal growth and dendritic solidification we present a numerical method which computes the motion of complex solid liquid boundaries in crystal growth the model we solve includes physical effects such as crystalline anisotropy surface tension molecular kinetics and undercooling the method is based on two ideas first the equations of motion are recast as a single history dependent boundary integral equation on the solid liquid boundary a fast algorithm is used to solve the integral equation efficiently second the boundary is moved by solving a hamiltonjacobi type equation on a fixed domain formulated by osher and sethian for a function in which the boundary is a particular level set this equation is solved by finite difference schemes borrowed from the technology of hyperbolic conservation laws the two ideas are combined by constructing a smooth extension of the normal velocity off the moving boundary in a way suggested by the physics of the problem our numerical experiments show the evolution of complex crystalline shapes development of large spikes and corners dendrite formation and side branching and pieces of solid merging and breaking off freely

a compressible mixture model with phase transition abstract we introduce a new thermodynamically consistent diffuse interface model of allen cahn navier stokes type for multi component flows with phase transitions and chemical reactions for the introduced diffuse interface model we investigate physically admissible sharp interface limits by matched asymptotic techniques we consider two scaling regimes i e a non dissipative and a dissipative regime where we recover in the sharp interface limit a generalized allen cahn euler system for mixtures with chemical reactions in the bulk phases equipped with admissible interfacial conditions the interfacial conditions satisfy for instance a young laplace and a stefan type law

dynamic solid solid transitions with phase characterized by an order parameter abstract we develop a thermodynamically consistent continuum theory for the study of solid solid phase transitions where deformation dominates heat and mass transfer we use an order parameter to characterize the notion of phase and identify phase interfaces with thin transition zones within which the strain and order parameter exhibit large gradients to model the growth of one phase at the expense of another we introduce configurational forces subject to their own balance that work against changes in the order parameter by studying traveling waves and the asymptotics of a transition layer we establish connections between our theory and more standard approaches that identify phase interfaces with surfaces of strain discontinuity

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

derivation of the wenzel and cassie equations from a phase field model for two phase flow on rough surface in this paper the equilibrium behavior of an immiscible two phase fluid on a rough surface is studied from a phase field equation derived from minimizing the total free energy of the system when the size of the roughness becomes small we derive the effective boundary condition for the equation by the multiple scale expansion homogenization technique the wenzel and cassie equations for the apparent contact angles on the rough surfaces are then derived from the effective boundary condition the homogenization results are proved rigorously by the convergence theory

dynamics of curved fronts in systems with power law memory abstract the dynamics of a curved front in a plane between two stable phases with equal potentials is modeled via two dimensional fractional in time partial differential equation a closed equation governing a slow motion of a small curvature front is derived and applied for two typical examples of the potential function approximate axisymmetric and non axisymmetric solutions are obtained

continuity of the temperature in the two phase stefan problem the classical two phase stefan problem a model for ice water melting gives rise to a singular nonlinear partial differential equation which admits a unique weak solution here we prove that this solution and therefore the temperature in the stefan problem are continuous

the dynamics of nucleation for the cahn hilliard equation when a constant metastable solution of the cahn hilliard equation is subjected to a spatially localized large amplitude perturbation a transition process may be triggered leading to a globally stable stationary solution in one space dimension the existence and instability of a third stationary solution with the same mass is proved a spike like solution called a canonical nucleus within the class of solutions which are even with respect to the center of the spike it has a one dimensional unstable manifold in addition the process of nucleation by formal arguments using two space scales and two timescales is described the last stage in the process can be approximated by a nonlinear stefan free boundary problem

conformal regularization of the kepler problem the manifoldm of null rays through the origin of 2 n 1 is diffeomorphic tos1 s n and it is a homogeneous space of so 2 n 1 this group therefore acts ont m which we show to be the generating manifold of the extended phase space of the regularized kepler problem a local canonical chart int m is found such that the restriction to the subbundle of the null nonvanishing covectors is given byp0 h q p 0 whereh q p is the hamiltonian of the kepler problem by means of this construction we get some results that clarify and complete the previous approaches to the problem

grain boundary motion arising from the gradient flow of the aviles giga functional abstract this paper considers the singular limit of the equation t 2 1 2 1 grain boundaries limiting discontinuities in form networks that coarsen over time a matched asymptotic analysis is used to derive a free boundary problem consisting of curve motion coupled along hyperbolic characteristics and junction conditions an intermediate boundary layer near extrema junctions is discovered along with the relevant nonlocal junction conditions the limiting dynamics can be viewed in the context of a gradient flow of the sharp interface energy on an attracting manifold dynamic scaling of the long time coarsening process can be explained by dimensional analysis of the reduced problem

thermodynamically consistent models of phase field type for the kinetics of phase transitions abstract a general framework is given for the phenomenological kinetics of phase transitions in which not only the order parameter but also the temperature may vary in time and space instead of a ginzburg landau free energy functional as used in formulating the cahn hilliard equation we use the analogous entropy functional model entropy functionals and the kinetic equations resulting from them are constructed for various cases phase transitions with and without a critical point and in the former case with or without a latent heat the class considered is general enough to include the entropy functionals for the ising model in mean field approximation the van der waals fluid and a simplified version of the density functional theory of freezing a case without critical point for which the energy is conserved but the order parameter is not provides a thermodynamically consistent derivation of the phase field equations studied by caginalp fix and others and also leads in a natural way to the lyapunov functional given by langer for these equations but the treatment also suggests that a modified version of the phase field equations might provide a more realistic model of freezing

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

sharp interface limits of the cahn hilliard equation with degenerate mobility in this work sharp interface limits for the degenerate cahn hilliard equation with a polynomial double well free energy and a mobility that vanishes at the minima of the double well are derived for the choice of a quadratic mobility the leading order sharp interface motion is not governed by pure surface diffusion as has been previously claimed in the literature but contains a contribution from nonlinear porous medium type bulk diffusion at the same order our analysis reveals that there are two subcases one where the solution for the order parameter is bounded between the minima proven to exist for the first mobility by elliott and garcke siam j math anal 27 1996 pp 404 423 and one where it converges to the classical stationary solution of the cahn hilliard equation consistent treatment of the bulk diffusion requires the matching of exponentially large and small terms in combination with multiple inner layers moreover the leading order sharp interface motion depends sensitively on the choice of mobility the asymptotic analysis shows that for example with a biquadratic mobility the leading order sharp interface motion is driven only by surface diffusion the sharp interface models are corroborated by comparing relaxation rates of perturbations to a radially symmetric stationary state with those obtained by the phase field model

phase field equations with memory the hyperbolic case we present a phenomenological theory for phase transition dynamics with memory which yields a hyperbolic generalization of the classical phase field model when the relaxation kernels are assumed to be exponential thereafter we focus on the implications of our theory in the hyperbolic case and we derive asymptotically an equation of motion in two dimensions for the interface between two different phases this equation can be considered as a hyperbolic generalization of the classical flow by mean curvature equation as well as a generalization of the born infeld equation we use a crystalline algorithm to study the motion of closed curves for the generalized hyperbolic flow by mean curvature equation our hyperbolic generalization of flow by mean curvature and present some numerical results which indicate that a certain type of two dimensional relaxation damped oscillation may occur

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

properties of some ordinary differential equations related to free boundary problems some second order ordinary differential equations of the form 2 2 n 1 r 3 k 0 are studied properties such as existence and monotonicity of solutions are considered for n 1 0 and two sets of boundary conditions for n 1 some explicit results are obtained for small these ode s arise from a phase field approach to free boundary problems involving a phase transition

shape instabilities and pattern formation in solidification a new method for numerical solution of the moving boundary problem abstract when a solidification front is advancing into a region of supercooled liquid its shape is subject to instabilities which can lead to complex modes of growth the evaluation of the growth pattern is determined by the interaction of the driving force of the instability due to heat diffusion with the restabilizing force due to surface tension and its full development is a highly nonlinear process a new method for solving the two phase heat diffusion problem under these conditions is described the method is based on a single domain treatment weak formulation of the moving boundary problem with the temperature field near the interface approximated by a piecewise linear function in the spirit of the finite element technique calculated results for a model problem are in excellent agreement with the predictions of an analytical approximation

on equations of motion on compact hermitian symmetric spaces the classical equations of motion on compact coherent state manifolds that have hermitian symmetric space structure are considered in the case of hamiltonians linear in the generators of the group that determines the structure of the manifold of coherent states the classical motion and the quantum evolution are both described by the same first order second degree differential equation the explicit forms of the classical equations of motion as matrix riccati equations on the grassmann manifold ga cn and the coset manifold so 2n u n attached to the hartree fock and respectively hartree fock bogoliubov problem are obtained

on the regularization of the kepler problem an intimate relationship between moser s regularization 1 and the ks regularization 3 of the 3 dimensional kepler problem is established explicit formulae linking moser s and the ks transformation are obtained in the case of negative as well as in the case of positive energies as a side result it is shown that the ks transformation owes its existence to the local isomorphism of so 2 4 and su 2 2

a multi phase mullins sekerka system matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem we propose a generalisation of the mullins sekerka problem to model phase separation in multi component systems the model includes equilibrium equations in bulk the gibbs thomson relation on the interfaces young s law at triple junctions together with a dynamic law of stefan type using formal asymptotic expansions we establish the relationship to a transition layer model known as the cahn hilliard system we introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds together with measure theoretical tools through an implicit time discretisation we construct approximate solutions by stepwise minimisation under the assumption that there is no loss of area as the time step tends to zero we show the existence of a weak solution

abstract approach to evolution equations of phase field type and applications a general system of abstract nonlinear parabolic equations deriving from phase field models of heat transfer in anisotropic fluids with phase transitions is studied through a backward finite differences approximation scheme and existence and uniqueness theorems are proved then applications are presented to two physical situations first a standard heat diffusion problem is briefly discussed mainly as a justification of the choices of the abstract setting second a more interesting transmission problem between two different adjoining fluids is studied under very general compatibility hypotheses and the abstract results are adapted to provide an existence and uniqueness theorem also in this case moreover through convex analysis techniques it is performed a mathematical study of the subdifferential type relations linking the phase fields and the temperatures to the internal energies of the fluids email schimper dragon ian pv cnr it keywords phase field models subdifferential operators abstract evolution equations

nonlinear preconditioning for diffuse interfaces a method of transforming problems with diffuse interfaces is presented which leads to equations that are easier to compute accurately information obtained by internal layer asymptotic analysis is utilized to motivate transformations of the dependent variables the new evolution equations which result from this change of variables can be solved numerically in a straightforward manner numerical experiments indicate that truncation errors can be significantly reduced in such problems allowing a coarser grid to be used applications to several well known models are presented

phase transitions in materials with thermal memory and a fourier term a model for thermally induced phase transitions in rigid materials with thermal memory was recently proposed both for the case where the phases have the same conductivity properties and where they are different in this work the model is generalized to the case of heatflow relations which include instantaneous contributions of the fourier type as well as memory terms the temperature gradient is decomposed into two parts each zero on one phase and equal to the temperature gradient on the other however they vary smoothly over the transition zone asymptotic analysis is carried out which shows that to leading order temperature is continuous across the transition zone and the normal derivatives of the temperature on each phase boundary obey a condition of the classical form with no explicit dependence on the memory terms this latter result emerges out of first order terms in the asymptotic analysis effects explicitly related to thermal memory only begin to play a role in the analysis of second order terms these results contrast sharply with those for materials without the instantaneous terms

review of some adaptive node movement techniques in finite element and finite difference solutions of partial differential equations abstract this article summarizes the results of a literature search for adaptive numerical methods of solving partial differential equations the methods discussed involve the adaptive movement of nodes so as to obtain a low level of solution truncation error while minimizing the number of nodes used in the calculation such methods are applicable to the solution of nonstationary flow problems that contain moving regions of rapid change in the flow variables surrounded by regions of relatively smooth variation flows with shock waves contact surfaces slip streams phase change interfaces and boundary layers can be modelled with great precision by these methods it will be shown that significant economies of execution can be attained if nodes are moved so that they remain concentrated in regions of rapid variation of the flow variables

dynamics of layered interfaces arising from phase boundaries the dynamics of a material in two phases is studied in the context of phase field models based on a landau ginzburg free energy functional they consist of a system of two nonlinear diffusion equat

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

a boundary conforming grid generation system for interface tracking abstract the development of an algebraic grid generation system to track a solid liquid interface during directional solidification of a binary alloy is discussed a single mapping constructed with tensor product b splines is proposed for calculations of both shallow and deep solidification cells the initial spline coefficients for the coordinate mapping are modified to minimize a discrete functional that regulates the smoothness and orthogonality of the mesh the use of transfinite blending function interpolation to obtain an initial grid is examined

a conserved phase field system with memory and relaxed chemical potential abstract in this paper we propose a conserved phase field model with memory and relaxed chemical potential our model could be considered also in the frame of cahn hilliard s theory of spinodal decomposition of a binary mixture where there is a coupling to a slowly relaxing variable e g in phase separation of polymer mixtures near a glass transition the global well posedness in an l p setting is obtained under suitable relaxation functions and under homogeneous conditions

an analysis of phase field alloys and transition layers a phase field approach to binary alloys is studied formal asymptotics of the system of parabolic differential equations leads to new interface relations as part of a macroscopic model which arises in the limit of vanishing interface thickness under suitable conditions we prove that the phase field system has a unique solution which converges to the limiting macroscopic solution the concentration and phase are monotonic across the interface for a simplified system transition layers in concentration are induced due to the change in phase and the change in material diffusion across the interface excess impurities may be trapped as a consequence of these layers

phase field models for hypercooled solidification abstract properties of the solidification front in a hypercooled liquid so called because the temperature of the resulting solid is below the melting temperature are derived using a phase field diffuse interface model certain known properties for hypercooled fronts in specific materials are reflected within our theories such as the presence of thin thermal layers and the trend towards smoother fronts with less pronounced dendrites when the undercooling is increased within the hypercooled regime both an asymptotic analysis to derive the relevant free boundary problems and a rigorous determination of the inner profile of the diffusive interface are given of particular interest is the incorporation of anisotropy and general microscale interactions leading to higher order differential operators these features necessitate a much richer mathematical analysis than previous theories anisotropic free boundary problems are derived from our models the simplest of which involves determining the evolution of a set a solid particle whose boundary moves with velocity depending on its normal vector considerable attention is given to the identification of surface tension to comparison with previous theories and to questions of stability

phase field approximations of the willmore functional and flow we discuss in this paper phase field approximations of the willmore functional and the associated mathrm l 2 l2 flow after recollecting known results on the approximation of the willmore energy and its mathrm l 1 l1 relaxation we derive the expression of the flows associated with various approximations and we show their behavior by formal arguments based on matched asymptotic expansions we introduce an accurate numerical scheme whose local convergence is proved to describe with more details the behavior of two flows the classical and the flow associated with an approximation model due to mugnai we propose a series of numerical simulations in 2d and 3d to illustrate their behavior in both smooth and singular situations

a microscopic derivation of macroscopic sharp interface problems involving phase transitions macroscopic free boundary problems involving phase transitions e g the classical stefan problem or its modifications are derived in a unified way from a hamiltonian based on a general set of microscopic interactions a hamiltonian of the form x x j x x x x leads to differential equations as a result of fourier transforms expanding the fourier transform ofj in powers ofq the wave number one can truncate the series at anarbitrary orderm and thereby obtainmth order differential equations an asymptotic analysis of these equations in various scalings of the physical parameters then implies limits which are the standard macroscopic models for the dynamics of phase boundaries

on anisotropic order parameter models for multi phase system and their sharp interface limits abstract for a general class of diffuse anisotropic multi phase order parameter or phase field models we use formally matched asymptotic expansions to determine the asymptotic limit when a small parameter related to the thickness of the interface tends to zero in the case of anisotropic allen cahn systems we obtain in the limit that the interface moves by anisotropic mean curvature flow at triple junctions a force balance holds which in the anisotropic case includes shear forces herring torque terms acting normal to the interface we further identify the singular limit of anisotropic cahn hilliard systems

efficient gluing of numerical continuation and a multiple solution method for elliptic pdes numerical continuation calculations for ordinary differential equations odes are by now an established tool for bifurcation analysis in dynamical systems theory as well as across almost all natural and engineering sciences although several excellent standard software packages are available for odes there are for good reasons no standard numerical continuation toolboxes available for partial differential equations pdes which cover a broad range of different classes of pdes automatically a natural ach to this problem is to look for efficient gluing computation approaches with independent components developed by researchers in numerical analysis dynamical systems scientific computing and mathematical modeling in this paper we shall study several elliptic pdes lane emden fowler lane emden fowler with microscopic force caginalp via the numerical continuation software pde2path and develop a gluing component to determine a set of starting solutions for the continuation by exploiting the variational structures of the pdes in particular we solve the initialization problem of numerical continuation for pdes via a minimax algorithm to find multiple unstable solution furthermore for the caginalp system we illustrate the efficient gluing link of pde2path to the underlying mesh generation and the fem matlab pdetoolbox even though the approach works efficiently due to the high level programming language and without developing any new algorithms we still obtain interesting bifurcation diagrams and directly applicable conclusions about the three elliptic pdes we study in particular with respect to symmetry breaking in particular we show for a modified lane emden fowler equation with an asymmetric microscopic force how a fully connected bifurcation diagram splits up into c shaped isolas on which localized pattern deformation appears towards two different regimes we conclude with a section on future software development issues that would be helpful to be addressed to simplify interfaces to allow for more efficient time saving gluing computation for dynamical systems analysis of pdes in the near future

a diffuse interface model for alloys with multiple components and phases a nonisothermal phase field model for alloys with multiple phases and components is derived the model allows for arbitrary phase diagrams we relate the model to classical sharp interface models by formally matched asymptotic expansions in addition we discuss several examples and relate our model to the ones already existing

on the evolution of domain walls in hard ferromagnets we propose a sharp interface theory for the dynamics of domain walls in highly anisotropic hard ferromagnetic bodies starting from the gilbert equation we consider the asymptotic regime when the hardness parameter goes to infinity and we use the technique of matched expansions to derive a system of two evolution equations for the domain wall regarded as a smooth surface the first equation apart for a nonlocal forcing term has the standard form for a surface set in motion according to its mean curvature the second relates the normal velocity to the internal structure of the domain wall

on a nonlocal phase field system abstract we study the existence uniqueness and continuous dependence on initial data of the solution to a nonlocal phase field system on a bounded domain the system is a gradient flow for a free energy functional with nonlocal interaction also we study the asymptotic behavior of the solution and show the existence of an absorbing set in some metric space