stochastic evolution of 2d crystals we study a crystalline version of the modified stefan problem in the plane the feature of our approach is that we consider a model with stochastic perturbations and assume the interfacial curve to be a polygon the existence of solution to our stochastic system is established galerkin s method is one of the main tools used in the proof of our assertion

retrieving topological information for phase field models the phase field approach has become a popular tool in modeling interface motion microstructure evolution and more recently the shape transformation of vesicle membranes under elastic bending energy while it is advantageous to employ phase field models in numerical simu lations to automatically handle topological changes to the microstructures or the configurations of vesicle membranes detecting topological events may also become important for many applications such as those in the simulation of blood cells motivated by such considerations a new quantity is formulated to retrieve some topological information based on the phase field formulation and to capture the occurrence of topological events it can also be used as a control method to avoid unphys ical changes of topology due to the numerical methods should it become necessary for particular practical applications through numerical experiments we demonstrate the effectiveness and the robustness of the new quantity in detecting the topology of fluid bubbles and vesicle membranes

well posedness results and asymptotic behaviour for a phase transition model taking into account microscopic accelerations we obtain an existence and uniqueness result to fremond s phase transition model which take into account microscopic movements and accelerations moreover the irreversible evolution of the phase variable is considered next we perform an asymptotic analysis on the solution to the above problem as the power of the microscopic acceleration forces goes to zero

on some geometric properties of generalized orlicz lorentz function spaces in this article we continue investigations concerning generalized orlicz lorentz function spaces initiated in the papers foralewski 2011 29 30 cf also foralewski 2008 27 28 first it is shown that modular is orthogonally subadditive next monotonicity properties are considered in order to get sufficient conditions for uniform monotonicity of the space a strong condition of 2 type and the notion of regularity of the generated musielak orlicz function are introduced finally criteria for non squareness of are presented

error estimates for a finite element discretization of a phase field model for mixtures we derive error estimates for finite element discretizations of phase field models that describe phase transitions in nonisothermal mixtures special attention is paid to the applicability of the result for a large class of models with nonlinear constitutive relations and to an approach that avoids an exponential dependence of the constants in the error estimate on the approximation parameter that models the thickness of the diffuse phase transition region the main assumptions on the model are a convexity condition for a function that can be interpreted as the negative local part of the entropy of the system a suitable regularity of the exact solutions and a spectrum estimate for the operator of the allen cahn equation the spectrum estimate is crucial to avoid the exponential dependence of error constants on the approximation parameters in the model this is done by a technique introduced in x feng and a prohl math comp 73 2004 pp 541 567 for phase transitions of pure materials with linear constitutive relations

martingale weak orlicz karamata hardy spaces associated with concave functions abstract in this paper we introduce the weak orlicz karamata hardy spaces associated with concave functions and establish the atomic decompositions as applications we prove several martingale inequalities and obtain a duality theorem

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

on the numerical solution of cauchy type singular integral equations by the collocation method the collocation method for the numerical solution of fredholm integral equations of the second kind is applied properly modified to the numerical solution of cauchy type singular integral equations of the first or the second kind but with constant coefficients this direct method of numerical solution of cauchy type singular integral equations is compared afterwards with the corresponding method resulting from applying the collocation method to the fredholm integral equation of the second kind equivalent to the cauchy type singular integral equation as well as with another method based also on the regularization procedure for the numerical solution of the same class of equations finally the convergence of the method is discussed

weak solutions to the stochastic porous media equation via kolmogorov equations the degenerate case a stochastic version of the porous medium equation with coloured noise is studied the corresponding kolmogorov equation is solved in the space l 2 h where is an infinitesimally excessive measure then a weak solution is constructed

stochastic porous media equations and self organized criticality the existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and wiener forcing term is proven the finite time extinction of solutions with high probability is also proven in 1 d the results are relevant for self organized criticality behavior of stochastic nonlinear diffusion equations with critical states

on a navier stokes allen cahn model with inertial effects abstract a mathematical model describing the flow of two phase fluids in a bounded container is considered under the assumption that the phase transition is influenced by inertial effects the model couples a variant of the navier stokes system for the velocity u with an allen cahn type equation for the order parameter relaxed in time in order to introduce inertia at the microscopic level the resulting model is characterized by second order material derivatives which constitute the main difficulty in the mathematical analysis in order to obtain a tractable problem a viscous relaxation term is then included in the phase equation the mathematical results consist in existence of weak solutions in 3d and under additional assumptions existence and uniqueness of strong solutions in 2d a partial characterization of the long time behavior of solutions is also given and in particular some issues related to dissipation of energy are discussed

kolmogorov equations in infinite dimensions well posedness and regularity of solutions with applications to stochastic generalized burgers equations we develop a new method to uniquely solve a large class of heat equations so called kolmogorov equations in infinitely many variables the equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper lyapunov type functions this way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally lipschitz drifts apart from general analytic interest the main motivation is to apply this to uniquely solve martingale problems in the sense of stroock varadhan given by stochastic partial differential equations from hydrodynamics such as the stochastic navier stokes equations in this paper this is done in the case of the stochastic generalized burgers equation uniqueness is shown in the sense of markov flows

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

weak solution of a stochastic 3d cahn hilliard navier stokes model driven by jump noise abstract we investigate a stochastic 2d and 3d cahn hilliard navier stokes system with a multiplicative noise of levy type the model consists of the navier stokes equations for the velocity coupled with a cahn hilliard system for the order phase parameter we prove that the system the existence of weak martingale solution for both 2d and 3d cases the proof of the existence is based on a classical galerkin approximation as well as some compactness methods in the 2d case we prove the pathwise uniqueness of the weak solution

the phase field method in the sharp interface limit the phase field pf method for solidification phenomena is an open formulation based on a free energy functional two common choices for the pf potential here referred to briefly as the caginalp and kobayashi models are compared with respect to their numeric results within the classical sharp interface limit both qualitative and quantitative behavior are addressed and an assessment of the computational effort required to approximate a sharp interface problem is made it is shown that the specific form of the free energy potential does have a strong influence on the convergence of the pf results to their sharp interface limit compliance of the pf solutions with the linear kinetic model for the interface temperature is also investigated a simple one dimensional solidification problem in the presence of kinetic undercooling is solved by the pf model and also by a deforming grid method our results support the view that if care is exercised in formulating the phase temperature coupling there is a high degree of confidence in using the pf method for the numerical modeling of general solidification phenomena

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

a variational approach to stochastic nonlinear parabolic problems abstract this work is concerned with an optimal control approach to stochastic nonlinear parabolic diffusion equations with monotonically increasing nonlinearity this approach leads to sharper existence and uniqueness results under minimal growth conditions on nonlinear diffusion coefficients

the gibbs thompson relation within the gradient theory of phase transitions this paper discusses the asymptotic behavior as 0 of the chemical potentials associated with solutions of variational problems within the van der waals cahn hilliard theory of phase transitions in a fluid with free energy per unit volume given by 2 2 w where is the density the main result is that is asymptotically equal to e d o with e the interfacial energy per unit surface area of the interface between phases the constant sum of principal curvatures of the interface and d the density jump across the interface this result is in agreement with a formula conjectured by m gurtin and corresponds to the gibbs thompson relation for surface tension proved by g caginalp within the context of the phase field model of free boundaries arising from phase transitions

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

reconstruction of two time independent coefficients in an inverse problem for a phase field system abstract in this paper we present stability results concerning the inverse problem of determining two time independent coefficients for a phase field system in a bounded domain r n for the dimension n 3 with a single observation on a subdomain and the sobolev norm of certain partial derivatives of the solutions at a fixed positive time 0 t over the whole spatial domain the proof of these results relies on an appropriate carleman estimate for the phase field system

quasilinear systems with memory kernels we consider a quasilinear integrodifferential system in non normal form such a system is a generalization of a phase field model with memory and includes as a particular case the system describing the combustion of a material with memory in this paper we study both the direct and the inverse problems our fundamental tools are the theory of analytic semigroups optimal regularity results and fixed point arguments

exponential attractors for a conserved phase field system with memory we consider a conserved phase field system of caginalp type characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient respectively the model consists of a parabolic integrodifferential equation coupled with a fourth order evolution equation for the phase field this system supplemented with suitable boundary conditions can be interpreted as a dissipative dynamical system in a proper phase space in a previous joint work the last two authors have proved the existence of a global attractor of finite fractal dimension here we show the existence of an exponential attractor by means of the techniques we developed for nonconserved phase field systems with memory

on the weighted galerkin method of numerical solution of cauchy type singular integral equations the well known weighted galerkin method for the direct numerical solution of cauchy type singular integral equations of the first and the second kind based on the approximation of the unknown function by a finite series of appropriate orthogonal polynomials is shown to be identical to the corresponding method based on the reduction of the cauchy type singular integral equation to an equivalent fredholm integral equation of the second kind and the application to the latter of the weighted galerkin method this result permits the direct transfer of the whole set of convergence and rate of convergence results for the weighted galerkin method of numerical solution of fredholm integral equations with regular kernels to the case of cauchy type singular integral equations an application to the case where chebyshev polynomials are used is also made and the available convergence results for the direct weighted galerkin method are rederived with essentially no analysis

atomic decompositions of vector valued weak musielak orlicz martingale spaces and applications abstract in this paper some vector valued weak martingale hardy spaces of musielak orlicz type are introduced and several atomic decompositions are established for them as applications of the atomic decompositions a sufficient condition for sublinear operators defined on vector valued weak musielak orlicz martingale spaces to be bounded is given using the sufficient condition some martingale inequalities are deduced these results closely depend on the geometrical properties of the banach space in which the martingales take values the results obtained here generalize the corresponding known results of scalar valued and vector valued weak martingale hardy spaces

embeddings between weak orlicz martingale spaces abstract let be an increasing and convex function on 0 with 0 0 satisfying that for any 0 there exists a positive constant c such that t c t t 0 let w l denote the corresponding weak orlicz space we obtain some embeddings between vector valued weak orlicz martingale spaces by establishing the w l inequalities for martingale transform operators with operator valued multiplying sequences these embeddings are closely related to the geometric properties of the underlying banach space in particular for any scalar valued martingale f f n n 1 we claim that sup n f n w l n 1 d f n 2 1 2 w l where d f n f n f n 1 and only depends on

the numerical computation of heteroclinic connections in systems of gradient partial differential equations the numerical computation of heteroclinic connections in partial differential equations pdes with a gradient structure such as those arising in the modeling of phase transitions is considered initially a scalar reaction diffusion equation is studied structural assumptions are made on the problem to ensure the existence of absorbing sets and consequently a global attractor as a result of the gradient structure it is known that if all equilibria are hyperbolic the global attractor comprises the set of equilibria and heteroclinic orbits connecting equilibria to one another thus it is natural to consider direct approximation of the set of equilibria and the connecting orbits results are proved about the fourier spanning basis for branches of equilibria and also for certain heteroclinic connections these results exploit the oddness of the nonlinearity the reaction diffusion equation is then approximated by a galerkin spectral discretization to produce a system of ordinary differential equations

finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise abstract in this paper one surveys and sharpens some recent results obtained by the authors on the finite time extinction property of solutions to stochastic diffusion equations of the form d x x m d t x d w and d x sgn x d t x d w where 0 m 1 0 these equations arise as models for nonlinear diffusion processes in porous media plasma and self organized criticality under stochastic gaussian perturbation these equations can be also viewed as control systems governed by fast diffusion porous media equations with a stochastic feedback controller x d w

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

the stability of the gauss chebyshev method for cauchy singular integral equations abstract we show that the infinity condition number of the gauss chebyshev method for the complete cauchy singular integral equation csie of the first kind is bounded above and below by the condition number of the dominant equation times a constant the condition number of the dominant equation is asymptotically equal to 0 718 0 344 ln n n 0 344 this implies that the gauss chebyshev method is stable for very large n s provided that the multiplying constants are not too large the magnitude of the constants depends on the eigenvalues of the csie

phase field method to optimize dielectric devices for electromagnetic wave propagation we discuss a phase field method for shape optimization in the context of electromagnetic wave propagation the proposed method has the same functional capabilities as the level set method for shape optimization the first advantage of the method is the simplicity of computation since extra operations such as re initialization of functions are not required the second is compatibility with the topology optimization method due to the similar domain representation and the sensitivity analysis structural shapes are represented by the phase field function defined in the design domain and this function is optimized by solving a time dependent reaction diffusion equation the artificial double well potential function used in the equation is derived from sensitivity analysis we study four types of 2d or 2 5d axisymmetric optimization problems two are the classical problems of photonic crystal design based on the bloch theory and photonic crystal wave guide design and two are the recent topics of designing dielectric left handed metamaterials and dielectric ring resonators

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

hardy lorentz spaces for b valued martingales abstract in this paper we study the hardy lorentz spaces for banach space valued martingales the dual spaces are characterized and several martingale inequalities are established the proofs mainly depend on the classical tool of atomic decompositions as usual these conclusions are closely related to the geometrical properties of the underlying banach spaces

hyperbolic phase field dynamics with memory abstract we consider a non conserved phase field system which consists of two nonlinearly coupled hyperbolic integrodifferential equations this model is derived from two basic assumptions the heat flux law is of gurtin pipkin type and the response of the order parameter to the variation of the free energy functional is delayed these hypotheses might be a reasonable attempt to describe for instance the melt of he 4 crystals a suitable initial and boundary value problem is then associated with the system and its well posedness is analyzed in detail

convergence and stability of a collocation method for the generalized airfoil equation we establish the mean square convergence and numerical stability of a collocation method for solving a class of cauchy singular integral equations the method uses the jacobi polynomials p 1 2 1 2 n as basis elements and the zeros of the polynomial p 1 2 1 2 n 1 as collocation points uniform convergence is shown to hold for the first iterate convergence is proved via vainikko s embedding method and stability is demonstrated by generalizing the approach of atkinson for integral equations of the second kind

computation of sharp phase boundaries by spreading the planar and spherically symmetric cases abstract the sharp interface that arises from any of the major transition problems classical or modified stefan etc can be smoothed using the phase field approach as a numerical tool the basic idea is that the thickness of the interface can be regarded as a mathematical free parameter which can be stretched beyond its physical value for computational convenience the computations in one dimensional space and n dimensions with radial symmetry indicate that this is an efficient method for dealing with stiff equations and results in a very accurate interface determination without explicit tracking the question of optimizing the interfacial thickness with respect to grid size is also considered empirically the technique also provides a numerical verification of the concept of an unstable critical radius of solidification

dual spaces of weak orlicz hardy spaces for vector valued martingales abstract the dual spaces of weak orlicz hardy spaces for b valued martingales which are defined by quasi concave functions are identified to obtain the results the so called weak type generalized campanato spaces are introduced and the classical technique of atomic decompositions is employed the results obtained here extend the corresponding known results in the literature from real valued setting to vector valued setting and from strong type spaces to weak type spaces

martingale transforms between orlicz hardy spaces of predictable martingales abstract using the technique of burkholder s martingale transforms the relations between predictable martingale orlicz hardy spaces are investigated let 1 and 2 be two young functions and 1 2 in some sense a constructive proof is obtained of that the elements in orlicz hardy space h 1 are none other than the martingale transforms of those in orlicz hardy space h 2 where h p q h s at the endpoint case the space h must be replaced by a b m o space it is also proved that a martingale is in h p q if and only if it is the transform of a martingale from b m o bmo 1 bmo 2

the two phase stochastic stefan problem this work is concerned with existence for a stochastic free boundary problem arising in phase transition the stefan two phase problem the existence of an invariant ergodic measure associated with the corresponding transition semigroup p t is also proved together with an integration by parts formula for the generator of p t

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