modelling phase transitions in alloys abstract we present a numerical method for solving a system of two nonlinear evolution equations of order four and two known as the cahn hilliard allen cahn system the simulation results give a first intuition about the stationary states form and stability

existence for coupled pseudomonotone strongly monotone systems and application to a cahn hilliard model with elasticity abstract a system of two operator equations is considered one of pseudomonotone type and the other of strongly monotone type both being strongly coupled conditions are given that allow to reduce the solvability of this system to a single operator equation of a pseudomonotone mapping this result is applied to a coupled system consisting of a parabolic equation of fourth order in space of cahn hilliard type and a nonlinear elliptic equation of second order modeling a quasi steady mechanical equilibrium using an appropriate notion of weak solutions and a framework for evolution equations developed by groger 2001 the system is reduced to a single parabolic operator equation and the existence of solutions is shown under restrictions on the strength of the coupling

modeling and analysis of a phase field system for damage and phase separation processes in solids abstract in this work we analytically investigate a multi component system for describing phase separation and damage processes in solids the model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function the main aim of this paper is to show existence of weak solutions for the introduced model where in contrast to existing damage models in the literature different elastic properties of damaged and undamaged material are regarded to prove existence of weak solutions for the introduced model we start with an approximation system then by passing to the limit existence results of weak solutions for the proposed model are obtained via suitable variational techniques

asymptotic stability of the stationary solution to an initial boundary value problem for the mullins equation of fourth order in the present article we first study the existence of the stationary solution to an initial boundary value problem for the mullins equation of fourth order which was proposed by mullins two dimensional motion of idealized grain boundaries j appl phys 27 900 1956 to describe the groove development due to the surface diffusion at the grain boundaries of a heated polycrystal then employing an energy method we prove that this stationary solution is asymptotically stable in a suitable norm as time goes to infinity

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

asymptotic evaluation of integrals related to time dependent fields near caustics integrals which arise in the theory of time dependent wave fields near caustics in nondispersive problems are much simpler than their time harmonic counterparts because they involve delta functions rather than exponentials fields near smooth caustics for instance involve solutions of cubic equations which can be expressed in elementary terms instead of the airy functions of the better known time harmonic theory the pearcey functions which arise in the theory of the cusped caustic are similarly replaced by expressions involving solutions of quartic equations etc as is well known each type of caustic is related to a particular type of catastrophe the theory presented here deals conveniently with fields reducible to single integrals since the delta function and the integral will annihilate each other leaving closed form expressions fields involving multiple integrals will be reduced somewhat in complexity but not in general to closed form and are not treated here

periodic solutions of non isothermal phase separation models with constraint abstract in this paper we consider the existence of time periodic solutions to the non isothermal phase separation models with constraints for the order parameter and the dirichlet boundary condition for the temperature we firstly address some approximate problems by the leray schauder fixed point theorem then the periodic solutions of the original problem are obtained by establishing some a priori estimates and performing a limiting procedure

a cahn hilliard equation with singular diffusion in the present work we address a class of cahn hilliard equations characterized by a singular diffusion term the problem is a simplified version with constant mobility of the cahn hilliard de gennes model of phase separation in binary incompressible isothermal mixtures of polymer molecules it is proved that for any final time t the problem admits a unique energy type weak solution defined over 0 t for any 0 such solution is classical in the sense of belonging to a suitable holder class over t and enjoys the property of being separated from the singular values corresponding to pure phases

convergence to equilibrium for the cahn hilliard equation with a logarithmic free energy abstract in this paper we investigate the asymptotic behavior of the nonlinear cahn hilliard equation with a logarithmic free energy and similar singular free energies we prove an existence and uniqueness result with the help of monotone operator methods which differs from the known proofs based on approximation by smooth potentials moreover we apply the lojasiewicz simon inequality to show that each solution converges to a steady state as time tends to infinity

evolution of phase boundaries by configurational forces an initial boundary value problem modeling the evolution of phase interfaces in materials showing martensitic transformations is studied the model which is derived rigorously from a sharp interface model with phase interfaces driven by configurational forces and which generalizes that model consists of the equations of linear elasticity coupled with a nonlinear partial differential equation of hyperbolic character governing the evolution of the order parameter it is proved that in one space dimension global solutions exist for which the order parameter belongs to the space of functions of bounded variation other models for interface motion by martensitic transformations and by interface diffusion are suggested

viscosity solutions to a new phase field model with neumann boundary condition for solid solid phase transitions abstract we shall study an initial boundary value problem of a new phase field model which is a degenerate parabolic equation coupled to a linear elasticity sub system used to describe the solid solid phase transitions in elastically deformable solid materials we establish a series of approximate solutions to the initial boundary value problem and prove that the viscosity solutions to this initial boundary value problem exist in a one dimensional case

the cahn hilliard equation with time dependent constraint abstract we propose an abstract variational inequality formulation of the cahn hilliard equation with a time dependent constraint we introduce notions of strong and weak solutions and prove that a strong solution if it exists is a weak solution and that the existence of a unique weak solution holds under an appropriate time dependence condition on the constraint we also show that the weak solution is a strong solution under appropriate assumptions on the data our abstract results can be applied to various concrete problems

a quasistatic evolution model for the interaction between fracture and damage this paper is devoted to the investigation of a quasistatic evolution model for a continuum which undergoes damage and possibly fracture in both cases the model appears to be ill posed so that it is necessary to introduce a relaxed variational evolution preserving the irreversibility of the process the minimality at each time and the energy balance from a mechanical point of view it turns out that the material prefers to form microstructures through the creation of fine mixtures between the damaged and healthy parts of the medium the brutal character of the damage process is then replaced by a progressive one where the original damage internal variable that is the characteristic function of the damaged part is replaced by the local volume fraction the analysis rests on a locality property for mixtures which enables us to use an alternative formula for the lower semicontinuous envelope of the elastic energy in terms of the g closure set

universal attractor for some singular phase transition systems a singular parabolic system describing the thermal diffusion in a substance possibly subject to a phase transition is introduced the physical process is described by the variables absolute temperature and order parameter the latter may have or not conserved total mass with respect to time in both cases after recalling and sometimes improving some known well posedness results the long time behavior of the system is studied it is shown that the process is dissipative and the compact universal attractor is constructed it turns out to attract the trajectories of the system in a rather strong metric which is strictly linked to the constraints imposed to both variables the techniques used in the proofs seem likely to be applied to other types of evolution systems containing maximal monotone nonlinearities

on nonoscillating integrals for computing inhomogeneous airy functions integral representations are considered of solutions of the inhomogeneous airy differential equation w z w pm1 pi the solutions of these equations are also known as scorer functions certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane by using steepest descent methods from asymptotics the standard integral representations of the scorer functions are modified in order to obtain non oscillating integrals for complex values of z in this way stable representations for numerical evaluations of the functions are obtained the methods are illustrated with numerical results

global existence and uniqueness for a singular degenerate cahn hilliard system with viscosity abstract existence and uniqueness are investigated for a nonlinear diffusion problem of phase field type consisting of a parabolic system of two partial differential equations complemented by neumann homogeneous boundary conditions and initial conditions this system aims to model two species phase segregation on an atomic lattice podio guidugli 2006 19 in the balance equations of microforces and microenergy the two unknowns are the order parameter and the chemical potential a simpler version of the same system has recently been discussed in colli et al 2011 8 in this paper a fairly more general phase field equation for is coupled with a genuinely nonlinear diffusion equation for the existence of a global in time solution is proved with the help of suitable a priori estimates in the case of a constant atom mobility a new and rather unusual uniqueness proof is given based on a suitable combination of variables

on a model for phase separation in binary alloys driven by mechanical effects this work is concerned with the mathematical analysis of a system of partial differential equations modeling the effect of phase separation driven by mechanical actions in binary alloys like tin lead solders the system combines the quasistationary balance of linear momentum with a fourth order evolution equation of cahn hilliard type for the phase separation and it is highly nonlinearly coupled existence and uniqueness results are shown tel 0382505631 email bonetti dimat unipv it

finiteness results for modular curves of genus at least 2 a curve x over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is modular if it is dominated by x 1 n for some n if in addition the image of its jacobian in j 1 n is contained in the new subvariety of j 1 n then x is called a new modular curve we prove that for each g 2 the set of new modular curves over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i of genus g is finite and computable for the computability result we prove an algorithmic version of the de franchis severi theorem similar finiteness results are proved for new modular curves of bounded gonality for new modular curves whose jacobian is a quotient of j 0 n new with n divisible by a prescribed prime and for modular curves new or not with levels in a restricted set we study new modular hyperelliptic curves in detail in particular we find all new modular curves of genus 2 explicitly and construct what might be the complete list of all new modular hyperelliptic curves of all genera finally we prove that for each field k of characteristic zero and g 2 the set of genus g curves over k dominated by a fermat curve is finite and computable

solutions to a model with nonuniformly parabolic terms for phase evolution driven by configurational forces we prove the existence of solutions global in time to an initial boundary value problem for a system of partial differential equations which consists of the equations of linear elasticity and a nonlinear nonuniformly parabolic equation of second order the problem models the behavior in time of materials with martensitic phase transformations this model with diffusive phase interfaces was derived from a model with sharp interfaces whose evolution is driven by configurational forces and can be considered to be a regularization of that model our existence proof which contributes to the verification of the model is only valid in one space dimension

regularity of the obstacle problem for the parabolic biharmonic equation we consider the obstacle problem for the parabolic biharmonic equation we study the problem via an implicit time discretization we prove the existence of a unique solution and discuss its regularity property

regularity of solutions to a model for solid solid phase transitions driven by configurational forces abstract in a previous work we prove the existence of weak solutions to an initial boundary value problem with h 1 initial data for a system of partial differential equations which consists of the equations of linear elasticity and a nonlinear degenerate parabolic equation of second order this problem models the behavior in time of materials with martensitic phase transitions this model with diffusive phase interfaces was derived from a model with sharp interfaces whose evolution is driven by configurational forces and can be regarded as a regularization of that model assuming in this article the initial data is in h 2 we investigate the regularity of weak solutions that is difficult due to the gradient term which plays a role of a weight our proof in which the difficulties are caused by the weight in the principle term is only valid in one space dimension

on the existence of solutions for nonlinear parabolic problems with nonmonotone discontinuities abstract in this paper we consider a nonlinear parabolic problem with a discontinuous nonmonotone nonlinearity we assume the existence of an upper solution and a lower solution such that using results from the theory of pseudomonotone operators and from the theory of multivalued analysis together with truncation and penalization techniques we show that there exists at least one solution x t z such that t z x t z t z a e on t z

on parity functions in conformal field theories we examine general aspects of parity functions arising in rational conformal field theories as a result of galois theoretic properties of modular transformations we focus more specifically on parity functions associated with affine lie algebras for which we give two efficient formulas we investigate the consequences of these for the modular invariance problem comment 18 pages no figure latex2e

aw1 p estimate for solutions to mixed boundary value problems for second order elliptic differential equations on demontre que sous des hypotheses plutot faibles une solution d un probleme aux valeurs limites mixte pour une equation elliptique d ordre 2 est dans l espace de sobolev w 1 p pour p 2

solvability via viscosity solutions for a model of phase transitions driven by configurational forces abstract this article is concerned with an initial boundary value problem for an elliptic parabolic coupled system arising in martensitic phase transition theory of elastically deformable solid materials e g steel this model was proposed in alber and zhu 2007 4 and investigated in alber and zhu 2006 3 the existence of weak solutions which are defined in a standard way however the key technique used in alber and zhu 2006 3 is not applicable to multi dimensional problem one of the motivations of this study is to solve this multi dimensional problem and another is to investigate the sharp interface limits thus we define weak solutions in a way which is different from alber and zhu 2006 3 by using the notion of viscosity solution we do prove successfully the existence of weak solutions in this sense for one dimensional problem yet the multi dimensional problem is still open

wave arrival singularities at cuspidal points in the acoustic wave surfaces of anisotropic solids and their unfolding under weak spatial dispersion this paper is concerned with wave arrival singularities in the elastodynamic green9s functions of infinite anisotropic elastic solids and their unfolding into smooth wave trains known as quasi arrivals through spatial dispersion the wave arrivals treated here are those occurring in i the displacement response to a suddenly applied point force or three dimensional green9s function and ii the displacement response to an impulsive line force or two dimensional green9s function these arrivals take on various analytical forms including step function and logarithmic and power law divergences they travel outwards from the source at the group velocities in each direction and their locus defines the three and two dimensional acoustic wave surfaces respectively the main focus of this paper is on the form of the wave arrivals in the neighbourhood of cuspidal points in the wave surfaces and how these arrivals unfold into quasi arrivals under the first onset of spatial dispersion this regime of weak spatial dispersion where the acoustic wavelength begins to approach the natural length scale l of the medium is characterized by a correction to the phase velocity which is quadratic in the wavevector k and the presence of fourth order spatial derivatives of the displacement field in the wave equation integral expressions are established for the quasi arrivals near to cuspidal points involving the airy function in the case of and the scorer function in the case of numerical results are presented illustrating the oscillatory nature of the quasi arrivals and the interference effects that occur near to cuspidal points in the wave surface

the nonlocal cahn hilliard equation with singular potential well posedness regularity and strict separation property abstract we consider the nonlocal cahn hilliard equation with singular potential and constant mobility well posedness and regularity of weak solutions are studied then we establish the validity of the strict separation property in dimension two further regularity results as well as the existence of regular finite dimensional attractors and the convergence of a weak solution to a single equilibrium are also provided finally regularity results and the strict separation property are also proven for the two dimensional nonlocal cahn hilliard navier stokes system with singular potential

the conserved penrose fife system with temperature dependent memory abstract a nonlinear parabolic system of penrose fife type with a singular evolution term arising from modelling dynamic phenomena of the nonisothermal diffusive phase separation is studied here we consider the evolution of a material in which the heat flux is a superposition of two different contributions one part is proportional to the spacial gradient of the inverse of the absolute temperature while the other agrees with the gurtin pipkin law introduced in the theory of materials with thermal memory the phase transition here is described through the evolution of the conserved order parameter which may represent the density or concentration of some substance it is shown that an initial boundary value problem for the resulting state equations has a unique solution