staggered lattice models some staggered models are studied along three different lines exact solvability yang baxter equations the existence of other exact solutions called disorder solutions their group of symmetries the analysis based on the commutation of row to row transfer matrices of arbitrary size n suggests that the exactly solvable models are either of non staggered type or of free fermion staggered type in contrast disorder solutions can easily be exhibited for staggered models the example of the ashkin teller model is analysed finally the symmetry group of these models is seen to be a straightforward generalisation of the one of homogeneous models

homogenization for semilinear hyperbolic systems with oscillatory data abstract the behavior of multi dimensional discrete boltzmann systems with highly oscillatory data is studied homogenized equations for the mean solutions are obtained uniform convergence of the oscillatory solutions of the discrete boltzmann equations to the solutions of the corresponding homogenized equations is established moreover we find that the weak limits of the oscillatory solutions for a model of broadwell type are not continuous functions of the discrete velocities generalization of the above results to problems with multiple scale initial data is also established reprints

homogenization of scalar conservation laws with oscillatory forcing terms the homogenization of a scalar conservation law with a highly oscillatory forcing term is studied effective equations are derived for the local averages of the oscillatory solutions together with approximations that correctly represent the phase and the amplitude of the oscillations a random choice method is also designed which as demonstrated by our numerical results gives the correct local averages without necessarily resolving the small scales finally the homogenization problem with an additional small viscosity term is studied

linearly localized difference schemes for the nonlinear maxwell model of a magnetic field into a substance abstract a linearly localized difference scheme with the first order time approximation is proposed for solving a nonlinear maxwell model associated with the penetration of a magnetic field into a substance the new scheme is computationally efficient since the resulting algebra equations are linear and can be computed by the fast thomas algorithm without any newton type inner iterations it is also local in time that is only numerical solutions in one previous time level are necessary to update the current solutions such that it requires much less storage compared with the fully implicit method furthermore the exponential decaying behavior of difference solution which is analogous to that of the continuous solution is obtained to improve the time accuracy we apply the crank nicolson type time discretization to construct a second order linearly localized method numerical examples are presented to support our theoretical results

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

numerical methods for oscillatory solutions to hyperbolic problems difference approximations of hyperbolic partial differential equations with highly oscillatory coefficients and initial values are studied analysis of strong and weak convergence is carried out in the practically interesting case when the discretization step sizes are essentially independent of the oscillatory wave lengths 01993 john wiley sons inc

on zamolodchikov s solution of the tetrahedron equations the tetrahedron equations arise in field theory as the condition for thes matrix in 2 1 dimensions to be factorizable and in statistical mechanics as the condition that the transfer matrices of three dimensional models commute zamolodchikov has proposed what appear from numerical evidence and special cases to be non trivial particular solutions of these equations but has not fully verified them here it is proved that they are indeed solutions

particle method approximation of oscillatory solutions to hyperbolic differential equations particle methods approximating hyperbolic partial diferential equations with oscillatory solutions are studied convergence is proved for approximations for which the continuous solution is not well resolved on the computational grid highly oscillatory solutions to the broadwell and variable coefficient carleman models are considered homogenization results are given and the approximations of more general systems are discussed numerical examples are presented

exponential decay of solutions of damped nonlinear hyperbolic equations on considere le comportement asymptotique des solutions de certaines equations hyperboliques non lineaires a amortissement lineaire

on the global solvability of kirchhoff equation for non analytic initial data abstract we give sufficient conditions for the global solvability of kirchhoff equation in terms of the spectral resolutions of the initial data u 0 x u t 0 x we assume no smallness conditions and only sobolev type regularity

hyperbolic parabolic singular perturbation for mildly degenerate kirchhoff equations time decay estimates we consider degenerate kirchhoff equations with a small parameter e in front of the second order time derivative it is well known that these equations admit global solutions when e is small enough and that these solutions decay as t with the same rate of solutions of the limit problem of parabolic type in this paper we prove decay error estimates for the difference between a solution of the hyperbolic problem and the solution of the corresponding parabolic problem these estimates show in the same time that the difference tends to zero both as e 0 and as t concerning the decay rates it turns out that the difference decays faster than the two terms separately as t proofs involve a nonlinear step where we separate fourier components with respect to the lowest frequency followed by a linear step where we exploit weighted versions of classical energies

factorized s matrices and generalized baxter models we prove that solutions of the factorization equations for s matrices with zn symmetry can be given in terms of theta functions we present a class of vertex models based on a symmetric s matrices a being an abelian group as natural generalizations of the baxter model the mapping of these models into spin systems is described

galerkin finite element method for one nonlinear integro differential model galerkin finite element method for the approximation of a nonlinear integro differential equation associated with the penetration of a magnetic field into a substance is studied first type initial boundary value problem is investigated the convergence of the finite element scheme is proved the rate of convergence is given too the decay of the numerical solution is compared with the analytical results

finite difference scheme for one system of nonlinear partial integro differential equations abstract system of maxwell equations is considered reduction to the integro differential form is given existence uniqueness and large time behavior of solutions of the initial boundary value problem for integro differential model with two component and one dimensional case are studied finite difference scheme is investigated wider class of nonlinearity is studied than one has been investigated before freefem realization code and results of numerical experiments are given

yosida approximation and nonlinear hyperbolic equation we are concerned with the initial value problem for nonlinear evolution equations of the form u t m a 1 2 u t 2 au t u t f t on 0 u 0 u o u o u 1 here a is a nonnegative selfadjoint operator in a real hilbert space h 0 is a constant and m is a c 1 class function satisfying m r m o 0 for r 0 with m o constant the equation has its motivation in the mathematical description of small amplitude vibrations of an elastic string

equivalence principle in statistical mechanics abstract we introduce the notion of equivalence of models of statistical mechanics which is stronger than the weak equivalence widely discussed in the literature we show that any generalized interaction round a face model is equivalent to a spin model and to a vertex model it implies in particular that for any spin model there is an equivalent vertex model

existence and regularity for a class of integrodifferential equations of parabolic type the method we use rests on the inversion of the formal laplace transform of r and allows one to analyze the regularity of the solution u problem 1 1 has been studied by several authors using various methods see for instance l 18 211 the laplace transform approach presented here has been used in s 7 9 14 153 in our previous papers s 7 93 we have in fact analyzed a more general situation while here we take full 36 0022 247x 85 3 00

regular solutions for time dependent abstract integrodifferential equations with singular kernel abstract results of existence uniqueness and regularity for strict and classical solutions of linear nonautonomous integrodifferential initial value problems are proved using maximal regularity properties for the autonomous nonintegral case

disorder solutions and the star triangle relation it is shown that the disorder solutions exhibited by two dimensional statistical lattice systems verify the star triangle relation these solutions of the star triangle relation are of a different type than those corresponding to completely integrable models

the initial value problem for measure valued solutions of a canonical 2 2 system with linearly degenerate fields we discuss serre s restricted formulation of the notion of a measure valued solution for a 2 2 system whose characteristic fields are both linearly degenerate we prove existence uniqueness and regularity results for this formulation as an application we prove the convergence of a particle method when the high frequency component of the exact solution is not resolved by the computational grid

on the abstract cauchy problem of parabolic type in spaces of continuous functions on etudie l existence l unicite et les proprietes de regularite des solutions du probleme de cauchy u t au r f t o t t u o u 0 ou a d a ce e est un operateur lineaire clos dans l espace de banach e

homogenization and convergence of the vortex method for 2 d euler equations with oscillatory vorticity fields the 2 d incompressible euler equations with oscillatory vorticity fields are studied a homogenization result for 2 d euler equations in velocity vorticity formulation is obtained and weak continuity of the equations is proved convergence of the vortex method is analyzed in the case when the continuous vorticity is not well resolved by the computational particles numerical results are given comparisons are made with the corresponding finite difference approximation

global existence and stability for wave equation of kirchhoff type with memory condition at the boundary abstract we consider a nonlinear wave equation of kirchhoff type with memory condition at the boundary and we study the asymptotic behavior of the corresponding solutions we proved that the energy decay with the same rate of decay of the relaxation function that is the energy decays exponentially when the relaxation function decay exponentially and polynomially when the relaxation function decay polynomially

tetrahedron equations and the relativistic s matrix of straight strings in 2 1 dimensions the quantums matrix theory of straight strings infinite one dimensional objects like straight domain walls in 2 1 dimensions is considered thes matrix is supposed to be purely elastic and factorized the tetrahedron equations which are the factorization conditions are investigated for the special two colour model the relativistic three strings matrix which apparently satisfies this tetrahedron equation is proposed

nonlinear parabolic equations with nonlinear functionals abstract in this paper we study a class of nonlinear parabolic equations in which certain functionals of the solution are involved a priori estimates for the solutions are derived by the use of integral estimates and the existence and uniqueness of the solutions are demonstrated via a retarding argument

a comment on baxter condition for commutativity of transfer matrices lett n andt n be the transfer matrices of two vertex models corresponding to two sets of boltzmann weights the baxter condition on boltzmann weights was known to be sufficient for commutativity oft n andt n for alln we show that generically it is also necessary

existence of global smooth solutions to the initial boundary value problem for the quasi linear wave equation with a degenerate dissipative term let c be a bounded domain with smooth boundary dq in the n dimen sional euclidean space rn and let us consider the initial boundary value problem for the classical quasi linear wave equation with a dissipation u v a lvul vu u x u o in qx o ce 4x 0 uo

necessary versus sufficient conditions for exact solubility of statistical models on lattices it is shown that under rather mild conditions the triangle relation represents a necessary condition for the existence of commuting transfer matrices of arbitrary size the cases of spin models and vertex models are treated separately

cα regularity for non autonomous linear integrodifferential equations of parabolic type abstract linear integrodifferential equations in general banach space are studied and applications are given to linear integrodifferential partial differential equations

let s baxterise we recall the concept of baxterisation of an r matrix or of a monodromy matrix which corresponds to build from one point in the r matrix parameter space the algebraic variety where the spectral parameter s live we show that the baxterisation which amounts to studying the iteration of a birational transformation is a win win strategy it enables to discard efficiently the non integrable situations focusing directly on the two interesting cases where the algebraic varieties are of the so called general type finite order iteration or are abelian varieties infinite order iteration we emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the baxter model we show that the baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying yang baxter structure is not used we baxterise l operators local quantum lax matrices and quantum hamiltonians

on a quasilinear wave equation with memory equation 1 l with a 0 and n 1 has its roots in a model for the small amplitude vibrations of a string in which the dependence of the tension on the deformation cannot be ignored see for example bernstein 2 carrier 3 41 dickey s and nishida 13 when a 0 1 1 may be used to describe the dynamics of an extensible string with fading memory this equation states that the dynamic equilibrium of a body depends not only on the present state of deformation but also on the previous history of the deformation gradient since the publication of the aforementioned papers equation 1 1 with a 0 has been the subject of intense scrutiny see for example arosio and spagnolo 11 lions 9 medeiros lo menzala ll nishihara 14 151 pohoiaev 16 and yamada 17 181 to name but a few among the facts thus far proved pohoiaev s results and their extension to the degenerate case of arosio and spagnolo are of particular significance they show that if the data satisfy an analytic type condition then solutions are global in time in spite of the abundant literature on the memoryless form of equation l l to our best knowledge equation 1 1 with a f 0 appears to be untouched our main results theorem 5 1 theorem 5 2 and theorem 5 4 will show that under appropriate assumptions on the kernel

a few remarks on the paper necessary versus sufficient conditions for exact solubility of statistical models on lattices j math phys 27 593 1986 lochak and maillard j math phys 27 593 1986 claim that the baxter condition which was known to be sufficient for the commutativity of transfer matrices is also necessary under some additional technical assumptions although the claim is correct the proof in that paper is false in this paper the errors of lochak and maillard are pointed out and correct proofs are outlined

automorphisms of algebraic varieties and yang baxter equations it is shown that commuting transfer matrix models in statistical mechanics are parametrized by algebraic varieties having a set of automorphisms deduced from the so called inversion relation in general this set of automorphisms is infinite this shows that for algebraic varieties of dimension 1 the models are parametrized by algebraic curves of genus 0 or 1

long time behavior of the two grid finite element method for fully discrete semilinear evolution equations with positive memory abstract based on two grid discretizations two fully discrete finite element algorithms for semilinear parabolic integro differential equations with positive memory are proposed with the backward euler scheme for the temporal discretization the basic idea of the space two grid finite element algorithms is to approximate the semilinear equations on a coarse space grid and to solve the linearized equations on a finer space grid at each time step to further decreases the amount of computational work a space time two grid algorithm based on a coarse space grid with large time stepsize t and a finer space grid with small time stepsize t for the evolutional equations is proposed in this paper the sharp long time stability and error estimates for the standard finite element method the space two grid finite element method and the space time two grid finite element method are derived it is showed that the two grid algorithms long time stability and error estimates are similar to those of the direct resolution of the semilinear problem on a fine grid