rotational degeneracy of hyperbolic systems of conservation laws the purpose of the paper is to study wave patterns of hyperbolic systems in which rotational symmetry creates a specific kind of degeneracy in this situation hyperbolicity is necessarily non strict so that the elementary waves have interesting patterns the discussion is centered around a theorem on existence and uniqueness of solutions of the riemann problem we give a unified presentation of examples from continuum mechanics

global smooth solutions for a class of quasilinear hyperbolic systems with dissipative terms in this paper we prove an existence theorem of global smooth solutions for the cauchy problem of a class of quasilinear hyperbolic systems with nonlinear dissipative terms under the assumption that only the c 0 norm of the initial data is sufficiently small while the c 1 norm of the initial data can be large the analysis is based on a priori estimates which are obtained by a generalised lax transformation

the riemann problem near a hyperbolic singularity ii this paper is interested in classifying the solutions of riemann problems for the 2 times 2 conservation laws that have homogeneous quadratic flux functions such flux functions approximate an arbitrary 2 times 2 system in a neighborhood of an isolated point where strict hyperbolicity fails here the solution for the symmetric systems in region iii of the four region classification of schaeffer and shearer is given the solution is based on the qualitative shape of the integral curves described by schaeffer and shearer and a numerical calculation of the hugoniot loci and their shock types

nonlinear stability of rarefaction waves for a viscoelastic material with memory in this paper we will discuss the stability of rarefaction waves for a viscoelastic material with memory the rarefaction waves for which the sta bility is tested are not themselves solutions to the integrodifferential equations 1 1 governing the viscoelastic material they are solutions to a related equi librium system of conservation laws given by 1 11 we shall show that if the forcing term and the past history are small and if the initial data are close to the rarefaction waves the solutions to 1 1 will approach the rarefaction waves in sup norm as the time goes to infinity j oo with the initial data u x 0 v x 0 u0 x v0 x xer l2 lim uo x v0 x u v x oo

the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

travelling wave solutions for a balance law we study entropy travelling wave solutions for first order hyperbolic balance laws results concerning existence regularity and asymptotic stability of such solutions are proved for convex fluxes and source terms with simple isolated zeros

a global formalism for nonlinear waves in conservation laws we introduce a unifying framework for treating all of the fundamental waves occurring in general systems ofn conservation laws fundamental waves are represented as pairs of states statisfying the rankine hugoniot conditions after trivial solutions have been eliminated by means of a blow up procedure these pairs form an n 1 dimensional manifold the fundamental wave manifold there is a distinguishedn dimensional submanifold of containing a single one dimensonal foliation that represents the rarefaction curves for all families similarly there is a foliation of itself that represent shock curves we identify othern dimensional submanifolds of that are naturally interpreted as boundaries of regions of admissible shock waves these submanifolds also have one dimensional foliations which represent curves of composite waves this geometric framework promises to simplify greatly the study of the stability and bifurcation properties

the classification of 2 2 systems of non strictly hyperbolic conservation laws with application to oil recovery on considere un systeme 2 2 a une variable d espace u f u x 0 0 ou u u x t r 2 et f r 2 r 2 on montre que pour des equations hyperboliques des points ombilicaux sont typiquement isoles et que en un point ombilical isole df est typiquement diagonalisable on donne une classification des equations hyperboliques a points ombilicaux isoles

higher order godunov methods for general systems of hyperbolic conservation laws abstract we describe an extension of higher order godunov methods to general systems of hyperbolic conservation laws this extension allow the method to be applied to problems that are not strictly hyperbolic and exhibit local linear degeneracies in the wave fields the method constructs an approximation of the riemann problem from local wave information a generalization of the engquist osher flux for systems is then used to compute a numerical flux based on this approximation this numerical flux replaces the godunov numerical flux in the algorithm thereby eliminating the need for a global riemann problem solution the additional modifications to the godunov methodology that are needed to treat loss of strict hyperbolicity are described in detail the method is applied to some simple model problems for which the global analytic structure is known the method is also applied to the black oil model for multiphase flow in petroleum reservoirs

asymptotic stability of rarefaction wave of the cauchy problem for viscous conservation laws abstract this paper is concerned with the asymptotic stability of rarefaction waves for the two dimensional steady isentropic irrotational flow with artificial viscosity we prove that if the initial value is close to a constant state and the corresponding inviscid hyperbolic system admits a weak rarefaction wave then the solution tends to this rarefaction wave as t

non parametric applications of an infinite dimensional convolution theorem this paper gives a general approach to the statistical problem of efficient non parametric estimation the main tool is an abstract infinite dimensional convolution theorem of the hajek le cam type this result is applied to the problems of efficiently estimating measures quantile functions spectral functions of stationary processes minimum distance functionals m functional and so forth

confidence sets for a multivariate distribution on etudie une version multivariable alternative de la bande de confiance unidimensionnelle de kolmogorov smirnov qui preserve l invariance affine

comparison of location models for stochastic processes let mathfrak e q be the statistical experiment based on the observation of an unknown function in the presence of an additive noise process with distributionq the possible loss of information whenq is replaced by some other noise distributionp is measured by the deficiency of mathfrak e p relative to mathfrak e q this deficiency and its relation to the variational distance ofp andq are studied mainly for gaussian noise processes gaussian diffusion processes and special set indexed processes are treated in detail

the global holder continuous solution of isentropic gas dynamics this paper considers the cauchy problem for the isentropic equations of gas dynamics in euler coordinates t u x 0 u t u 2 p x 0 and gives the holder continuous solution by applying the method of vanishing viscosity

finite lattice extrapolations for z3 and zs models a method is presented to accelerate the convergence of finite lattice sequences to their bulk limit the calculation of highly accurate estimates of the critical parameters of the bulk system is then possible applied to the hamiltonian version of the z3 model three state potts model in 1 1 dimensions these techniques yield estimates for the exponents gamma 1 444 or 0 0001 nu 0 8333 or 0 0003 and alpha 0 33 or 0 01 for the z5 model the presence of a kosterlitz thouless transition is confirmed

nonlinear stability of an undercompressive shock for complex burgers equation though there is strong numerical evidence for the stability of undercompressive shocks their stability has not been verified analytically in particular the energy methods used to analyze stability of standard shocks do not apply here we present the first proof of stability for a particular undercompresive shock the real burgers shock considered as a solution of the complex burgers equation our analysis is by direct calculation of the green s function for the linearized equations combined with pointwise estimates of nonlinear effects a benefit of this method is to obtain fairly detailed information about the solution includingl1 behavior and rates of decay in different regions of space

existence and asymptotic behavior of solutions to inhomogeneous equations of elasticity with little viscosity we study the existence and asymptotic behavior of the solutions to inhomogeneous equations of elasticity with little viscosity u t x v f u v eu xx v t u x g u v ev xx with initial value u v x 0 u 0 x v 0 x here v is a given smooth strongly monotone function on r v a 2 v a v 0 ver and f u v and g u v are given smooth functions on r 2 and e is a positive constant

nonlinear stability of rarefaction waves for compressible navier stokes equations it is shown that expansion waves for the compressible navier stokes equations are nonlinearly stable the expansion waves are constructed for the compressible euler equations based on the inviscid burgers equation our result shows that navier stokes equations and euler equations are time asymptotically equivalent on the level of expansion waves the result is proved using the energy method making essential use of the expansion of the underlining nonlinear waves and the specific form of the constitutive eqution for a polytropic gas

degenerate hyperbolic conservation laws with dissipation reduction to and validity of a class of burgers type equations a conservation law is said to be degenerate or critical if the jacobian of the flux vector evaluated on a constant state has a zero eigenvalue in this paper it is proved that a degenerate conservation law with dissipation will generate dynamics on a long time scale that resembles burger s dynamics the case of k fold degeneracy is also treated and it is shown that it leads to a reduction to a quadratically coupled k fold system of burgers type equations validity of the reduction and existence for the reduced system is proved in the class of uniformly local spaces thereby capturing both finite and infinite energy solutions the theory is applied to some examples from stratified shallow water hydrodynamics that model the birth of hydraulic jumps

one dimensional ising models with long range interactions we consider ising models with long range ferromagnetic pair interactions decaying as for we first find approximate values for the critical temperature we use a cluster mean field approach combined with finite size scaling and vanden broeck and schwartz transformations for we find which can be compared with recent results of luijten and blote who found and which is two orders of magnitude more accurate than any previous results since we use a mean field cluster approximation as part of our approach the accuracy for larger values of decreases significantly in addition to we obtain approximate values for the critical exponents and using the coherent anomaly method for we obtain and all extremely close to the predictions of renormalization group calculations which say that these exponents should take on their classical values for this value of

asymptotic stability of rarefaction waves for 2 2 viscous hyperbolic conservation laws the two modes case abstract in this paper we continue our study on the asymptotic behavior toward rare faction waves of a general 2 2 system of hyperbolic conservation laws with positive viscosity matrix it is shown that when the initial data is a small perturbation of a weak rarefaction wave a linear superposition of a 1 rarefaction wave and a 2 rarefaction wave for the corresponding inviscid hyperbolic conservation laws then the solution of the cauchy problem for the viscous system globally exists and tends to the rarefaction wave the result is proved by using an energy method combining the technique in z p xin j differential equations 73 1988 45 77 and using the characteristic energy method of t p liu mem amer math soc 328 1975 1 108

on nonlinear stability of general undercompressive viscous shock waves we study the nonlinear stability of general undercompressive viscous shock waves previously the authors showed stability in a special case when the shock phase shift can be determined a priori from the total mass of the perturbation using new pointwise methods by examining time invariants associated with the linearized equations we can now overcome a new difficulty in the general case namely nonlinear movement of the shock we introduce a coordinate transformation suitable to treat this new aspect and demonstrate our method by analyzing a model system of generic type we obtain sharp pointwise bounds andlp behavior of the solution for allp 1 p

long range one dimensional potts models a cluster mean field and extrapolation approach we obtain approximations for the critical temperature tc of q state ferromagnetic potts models on one dimensional lattices with algebraically decaying ferromagnetic pair interactions i e decaying as 1 r with 1 2 initially we use a cluster mean field method to get approximations to the critical temperature and we look at increasingly large clusters this gives us a sequence of ever more accurate approximate values which can be used as input to various extrapolation algorithms we see that as the interaction decreases more slowly and the value of q increases we obtain more and more accurate tc estimates our best estimates we believe to be correct to four figure accuracy

initial boundary value problem for conservation laws this paper concerns the initial boundary value problems for some systems of quasilinear hyperbolic conservation laws in the space of bounded measurable functions the main assumption is that the system under study admits a convex entropy extension it is proved that then any twicely differentiable entropy fluxes have traces on the boundary if the bounded solutions are generated by either godunov schemes or by suitable viscous approximations furthermore in the case that the weak interior solutions are generated by godunov schemes any lipschitz continuous entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boundary values this in particular gives a trace formula for the flux functions in terms of the numerical boundary data we also investigate the formulation of boundary conditions for systems of hyperbolic conservation laws it is shown that the set of expected boundary values derived from the viscous approximation contains the one derived in terms of the boundary riemann problems and the converse is not true in general the general theory is then applied to some specific examples first several new facts are obtained for convex scalar conservation laws for example we give example which show that godunov schemes produce numerical boundary layers it is shown that any continuous functions of density have traces on the boundary instead of only entropy fluxes we also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data a generalized oleinik entropy condition is also obtained next we prove the existence of a weak solution to the initial boundary value problem for a family of quadratic system with a uniformly characteristic boundary condition

a higher order godunov method for modeling finite deformation in elastic plastic solids in this paper we develop a first order system of conservation laws for finite deformation in solids describe its characteristic structure and use this analysis to develop a second order numerical method for problems involving finite deformation and plasticity the equations of mass momentum and energy conservation in lagrangian and eulerian frames of reference are combined with kinetic equations of state for the stress and with caloric equations of state for the internal energy as well as with auxiliary equations representing equality of mixed partial derivatives of the deformation gradient particular attention is paid to the influence of a curl constraint on the deformation gradient so that the characteristic speeds transform properly between the two frames of reference next we consider models in rate form for isotropic elastic plastic materials with work hardening and examine the circumstances under which these models lead to hyperbolic systems for the equations of motion in spite of the fact that these models violate thermodynamic principles in such a way that the acoustic tensor becomes nonsymmetric we still find that the characteristic speeds are always real for elastic behavior and essentially always real for plastic response these results allow us to construct a second order godunov method for the computation of three dimensional displacement in a one dimensional material viewed in the lagrangian frame of reference we also describe a technique for the approximate solution of riemann problems in order to determine numerical fluxes in this algorithm finally we present numerical examples of the results of the algorithm

multiple mode diffusion waves for viscous nonstrictly hyperbolic conservation laws we study the large time behaviors of solutions of viscous conservation laws whose inviscid part is a nonstrictly hyperbolic system the initial data considered here is a perturbation of a constant state it is shown that the solutions converge to single mode diffusion waves in directions of strictly hyperbolic fields and to multiple mode diffusion waves in directions of nonstrictly hyperbolic fields the multiple mode diffusion waves which are the new elements here are the self similar solutions of the viscous conservation laws projected to the nonstrictly hyperbolic fields with the nonlinear fluxes replaced by their quadratic parts the convergence rate to these diffusion waves iso t 3 4 1 2p inl p 1 p with 0 being arbitrarily small

notes on the eigensystem of magnetohydrodynamics the eigenstructure of the equations governing one dimensional ideal magnetohy drodynamics is examined motivated by the wish to exploit it for construction of high resolution computational algorithms the results are given in simple forms that avoid indeterminacy or degeneracy whenever possible the unavoidable indeterminacy near the magnetosonic or triple umbilic state is analysed and shown to cause no difficulty in evaluating a numerical flux function the structure of wave paths close to this singularity is obtained and simple expressions are presented for the structure coefficients that govern wave steepening

global stability of viscous contact wave for 1 d compressible navier stokes equations the viscous contact waves for one dimensional compressible navier stokes equations has recently been shown to be asymptotically stable the stability results are called local stability or global stability depending on whether the norms of initial perturbations are small or not up to now local stability results toward viscous contact waves of compressible navier stokes equations have been well established see huang et al 2006 2008 2009 9 10 7 but there are few results for the global stability in the case of cauchy problem which is the purpose of this paper the proof is based on an elementary energy method using an inequality concerning the heat kernel see lemma 1 of huang et al 2010 7

fast flat histogram method for generalized spin models we present a monte carlo method that efficiently computes the density of states for spin models having any number of interaction per spin by combining a random walk in the energy space with collective updates controlled by the microcanonical temperature our method yields dynamic exponents close to their ideal random walk values reduced equilibrium times and very low statistical error in the density of states the method can host any density of states estimation scheme including the wang landau algorithm and the transition matrix method our approach proves remarkably powerful in the numerical study of models governed by long range interactions where it is shown to reduce the algorithm complexity to that of a short range model with the same number of spins we apply the method to the q state potts chains 3 ensuremath leqslant q ensuremath leqslant 12 with power law decaying interactions in their first order regime we find that conventional local update algorithms are outperformed already for sizes above a few hundred spins by considering chains containing up to 2 16 spins which we simulated in fairly reasonable time we obtain estimates of transition temperatures correct to five figure accuracy finally we propose several efficient schemes aimed at estimating the microcanonical temperature

asymptotic stability of monotone increasing traveling wave solutions for viscous compressible fluid equations with capillarity term abstract asymptotic stability of monotone increasing traveling wave solutions is investigated for a class of viscous compressible fluid equations with capillarity term by using the theory of planar dynamical systems the existence has been proved for the monotone increasing traveling wave solution moreover through building uniformly a priori energy estimate for the perturbation of the traveling wave solution we prove that the monotone increasing traveling wave solution is asymptotically stable in h 2 h 1

decay rates of solutions to the cauchy problem for dissipative nonlinear evolution equations abstract in this paper we study the global existence and the asymptotic behavior of the solutions to the cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects e t 1 x x x t 1 x x x x with initial data i x 0 0 x 0 x as x where and are positive constants such that 1 1 through constructing a correct function x t defined by 2 13 and using the energy method we show sup x r x t x x x t 0 as t and the solutions decay with exponential rates the same problem was studied by tang and zhao s q tang h j zhao nonlinear stability for dissipative nonlinear evolution equations with ellipticity j math anal appl 233 1999 336 358 for the case of 0 0

large time behaviors of solutions to an inflow problem in the half space for a one dimensional system of compressible viscous gas the inflow problem for a one dimensional compressible barotropic flow on the half line r 0 is investigated not only classical waves but also the new wave which is called the boundary layer solution arise large time behaviors of the solutions to be expected have been classified in terms of the boundary values by a matsumura inflow and outflow problems in the half space for a one dimensional isentropic model system of compressible viscous gas to appear in proceedings of ims conference on differential equations from mechanics hong kong 1999 in this paper we give the rigorous proofs of the stability theorems on both the boundary layer solution and a superposition of the boundary layer solution and the rarefaction wave

nonlinear stability of strong rarefaction waves for the generalized kdv burgers kuramoto equation with large initial perturbation abstract in was shown in ruan et al 2008 3 that rarefaction waves for the generalized kdv burgers kuramoto equation are nonlinearly stable provided that both the strength of the rarefaction waves and the initial perturbation are sufficiently small the main purpose of this work is concerned with nonlinear stability of strong rarefaction waves for the generalized kdv burgers kuramoto equation with large initial perturbation in our results we do not require the strength of the rarefaction waves to be small and when the smooth nonlinear flux function satisfies certain growth condition at infinity the initial perturbation can be chosen arbitrarily in h 1 r while for a general smooth nonlinear flux function we need to ask for the l 2 norm of the initial perturbation to be small but the l 2 norm of the first derivative of the initial perturbation can be large and consequently the h 1 r norm of the initial perturbation can also be large

non classical riemann solvers with nucleation we introduce a new nonclassical riemann solver for scalar conservation laws with concave convex flux function this solver is based on both a kinetic relation which determines the propagation speed of undercompressive nonclassical shock waves and a nucleation crite rion which makes a choice between a classical riemann solution and a nonclassical one we establish the existence of nonclassical entropy solutions of the cauchy problem and discuss several examples of wave interactions we also show the existence of a class of solutions called splitting merging solutions which are made of two large shocks and small bv bounded variation perturbations the nucleation solvers as we call them are applied to and actually motivated by the theory of thin film flows they help explain numerical results observed for such flows

effective dimensions in networks with long range connections abstract one and two dimensional lattices of points are connected with long range links whose lengths are distributed according to p r r by changing the decay exponent one can go from d dimensional short range networks to dimensional networks topologically similar to random graphs an effective dimension d chem can be defined in terms of the shortest path properties of these networks these effective dimensions d chem are calculated here in one and two dimensions for system sizes of up to 10 7 points

the riemann problem for 2 2 systems of hyperbolic conservation laws with case i quadratic nonlinearities on etudie les problemes de riemann pour des systemes 2 2 de lois de conservation non strictement hyperboliques avec des non linearites quadratiques

asymptotic stability of traveling wave solutions of systems for one dimensional gas motion the asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics 1 the solution of a scalar conservation law with viscosity approaches the traveling wave solution at the ratet for some 0 ast provided that the initial disturbance is small and of integral zero and in addition decays at an algebraic rate for x 2 the traveling wave solution with nishida and smoller s condition of the system of a viscous heat conductive ideal gas is asymptotically stable provided the initial disturbance is small and of integral zero 3 the traveling wave solution with weak shock profile of the broadwell model system of the boltzmann equation is asymptotically stable provided the initial disturbance is small and its hydrodynamical moments are of integral zero each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one the property of integral zero of the initial disturbance plays a crucial role in this procedure

riemann problems for nonstrictly hyperbolic 2 2 systems of conservation laws on resout le probleme de riemann pour des systemes 2 2 de lois de conservation hyperboliques ayant des fonctions flux quadratiques

the interaction of nonlinear hyperbolic waves nonlinearities in wave equations lead to focusing and defocusing of solutions focusing causes sharply defined wave fronts the interaction of such sharply defined wave fronts and more generally of nonlinear hyperbolic waves is of fundamental importance and includes such phenomena as mach triple point formation shock wave diffraction patterns and the study of riemann problems in one and higher dimensions recent progress in the study of nonlinear hyperbolic wave interactions has revealed a surprising range of new mathematical phenomena and structures this mathematical theory should be useful in the design of improved computational algorithms and in part was motivated by such considerations it is also of considerable interest for its own sake as new mathematical phenomena as well as in terms of the direct insight it provides into physical phenomena within the subject matter and point of view adopted here we have attempted to present a broad and we hope a representative account of recent progress

comparing location experiments in sections 1 3 the classical theory of the comparison of two experiments is reviewed with particular reference to the comparison of two location experiments it is shown that the requirement of domination of one experiment by another for all decision problems is too strong to provide a reasonable basis for comparison for one parameter problems with monotone likelihood ratio it is therefore proposed to restrict the comparison to decision problems that are monotone in the sense of karlin and rubin 1956 application of this weaker definition to the location problem is shown to give satisfactory results a scale free comparison of this type leads to a new tail ordering of distributions and this is explored in section 6

hyperbolic conservation laws with umbilic degeneracy i in this paper a compactness framework for approximate solutions to nonlinear hyperbolic systems with umbilic degeneracy is established by combining techniques of compensated compactness with some classical methods and by a detailed analysis of a highly singular equation of euler poisson darboux type then this framework is successfully applied to prove the convergence of the viscosity method and to prove the existence of global entropy solutions for the cauchy problem with large initial data for a canonical class of the systems with quadratic flux form

stability of rarefaction waves in viscous media we study the time asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one dimensional viscous media with strictly hyperbolic flux functions our main result is to show that solutions of perturbed rarefaction data converge to an approximate burgers rarefaction wave for initial perturbations w 0 with small mass and localized as w 0 x mathcal o x 1 the proof proceeds by iteration of a pointwise ansatz for the error using integral representations of its various components based on green s functions we estimate the green s functions by careful use of the hopf cole transformation combined with a refined parametrix method as a consequence of our method we also obtain rates of decay and detailed pointwise estimates for the error this pointwise method has been used successfully in studying stability of shock and constant state solutions new features in the rarefaction case are time varying coefficients in the linearized equations and error waves of unbounded mass mathcal o log t these diffusion waves have amplitude mathcal o t 1 2logt in linear degenerate transversal fields and mathcal o t 1 2logt in genuinely nonlinear transversal fields a distinction which is critical in the stability proof

asymptotic stability of rarefaction waves for the generalized kdv burgers equation on the half line abstract we investigate the asymptotic behavior of solutions of the initial boundary value problem for the generalized kdv burgers equation u t f u x u x x u x x x on the half line with the boundary condition u 0 t u the corresponding cauchy problems of the behaviors of weak and strong rarefaction waves have respectively been studied by wang and zhu z a wang c j zhu stability of the rarefaction wave for the generalized kdv burgers equation acta math sci 22b 3 2002 309 328 and duan and zhao r duan h j zhao global stability of strong rarefaction waves for the generalized kdv burgers equation nonlinear anal tma 66 2007 1100 1117 in the present problem on the basis of the dirichlet boundary conditions the asymptotic states are divided into five cases dependent on the signs of the characteristic speeds f u in the cases of 0 f u f u we prove the global existence of solutions and asymptotic stability of the weak rarefaction waves when the initial disturbance is small also we can get asymptotic stability of the strong rarefaction waves when f u satisfies a certain growth condition

asymptotic stability of rarefaction waves for 2 2 viscous hyperbolic conservation laws this paper concerns the asymptotic behavior toward rarefaction waves of the solution of a genera 2 x 2 hyperbolic conservation laws with positive viscosity we prove that if the initial data is close to a constant state and its values at a3 lie on the kth rarefaction curve for the corresponding hyperbolic conservation laws then the solution tends as i co to the rarefaction wave determined by these states

positively invariant regions for a problem in phase transitions positively invariant regions for the system v t p w x e v xx w t v x e w x are constructed where p p w 0 w e 0 such a choice of p is motivated by the maxwell construction for a van der waals fluid the method of an analysis is a modification of earlier ideas of chueh conley smoller 1 the results given here provide e independent l bounds on the solution w v

adaptive finite element methods for conservation laws based on a posteriori error estimates we prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension and design corresponding adaptive methods the proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem together with the galerkin orthogonality of the finite element method the strong stability estimate uses the entropy condition for the system in an essential way 1995 john wiley sons inc

global stability of strong rarefaction waves for the generalized kdv burgers equation this paper is concerned with the global stability of strong rarefaction waves for the generalized kdv burgers equation in contrast to former results obtained by z a wang and c j zhu stability of the rarefaction wave for the generalized kdv burgers equation acta math sci 22b 3 2002 319 328 ours do not require the strength of the rarefaction waves to be small and the initial disturbance can also be chosen large c 2006 elsevier ltd all rights reserved

the coherent anomaly method and long range one dimensional ising models the authors present the results of the coherent anomaly method when applied to ising models in one dimension with long range interactions this class of systems acts as an interesting and challenging test for the method because the critical exponents as well as the critical temperature values of which are given by the method depend on the rate of fall off of the interactions thus one can see how accurately the method gives results which correctly reflect this dependency the results obtained from this method are compared with results obtained by a variety of other methods

the jin xin relaxation approximation of scalar conservation laws in several dimensions with large initial perturbation abstract this paper is concerned with nonlinear stability of strong planar rarefaction waves for the jin xin relaxation approximation of scalar conservation laws in several dimensions for such a problem local stability of weak or strong planar rarefaction waves have been obtained in luo 1997 20 and zhao 2000 43 respectively for the global stability results to the best of our knowledge the only result available now is on the one dimensional case cf zhao 2000 43 which is based on the maximum principle established in natalini 1996 30 the main purpose of this paper is try to deduce some nonlinear stability results with large initial perturbation our analysis is based on the elementary energy method and the continuation argument

weak stability in the global norm for systems of hyperbolic conservation laws we prove that solutions for systems of two conservation laws which are generated by glimm s method are weakly stable in the global l norm the method relies on a previous decay result of the author together with a new estimate for the l1 lipschitz constant that relates solutions at different times the estimate shows that this constant can be bounded by the supnorm of the solution and is proved for any number of equations the techniques do not rely on the existence of a family of entropies and moreover the results would generalize immediately to more than two equations if one were to establish the stability of solutions in the supnorm for more than two equations

asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions this paper concerns the large time behavior toward planar rarefaction waves of the solutions for scalar viscous conservation laws in several dimensions it is shown that a planar rarefaction wave is nonlinearly stable in the sense that it is an asymptotic attractor for the viscous conservation law this is proved by using a stability result of rarefaction wave for scalar viscous conservation laws in one dimension and an elementary l 2 energy method 0 introduction we will establish the asymptotic stability of planar rarefaction waves for scalar viscous conservation laws in two or more space dimensions we consider n dimensional scalar viscous conservation laws of the form n n 1 ut el f u x e aijux1u x e r t 0 i 1 i j l where u e r1 a aij called the viscosity matrix is a constant positive definite matrix and we assume that all the flux functions are smooth say in cn and equation 1 is genuinely nonlinear in the x1 direction 8 i e for a fixed constant a 0 2 jf u a the initial data for equation 1 is 3 u x o uo x satisfying 4 lim u x1 u ilx rn i 0 where u u u are two constants a planar rarefaction wave in x1 direction ur xl t is a solution of the following initial value problem for the received by the editors september 27 1988 1980 mathematics subject classification 1985 revision primary 35q99 35k35

a numerical study of riemann problem solutions and stability for a system of viscous conservation laws of mixed type a numerical study of the isothermal fluid equations with a nonmonotone equation of state like that of van der waals and with viscosity and capillarity terms is presented this system is ill posed i e elliptic in x vs t in some regions of state space and well posed i e hyperbolic in other regions thus it may serve as a model for describing dynamic phase transitions numerical computations of phase jumps shock waves and rarefaction waves for this system are presented although the solution of the riemann problem is not unique all of these waves are found to be stable to infinitesimal initial perturbations criteria are found for instability after o 1 initial perturbations an analytic argument is made for stability of phase transitions

viscous solutions of quadratic conservation laws with umbilic points viscous solutions of quadratic conservation laws with umbilic points ltj yunguang zhu changjiang and zhao huijiang wuhan institute of mathematical sciences academia sinica wuhan 430071 people s republic of china received 10 january 1991 received in revised form 4 august 1992 received for publication 1 march 1993

decay of entropy solutions of nonlinear conservation laws we are concerned with the asymptotic behavior of entropy solutions of nonlinear conservation laws the main objective of this paper is to present an analytical approach and to explore its applications to studying the large time behavior of periodic entropy solutions of hyperbolic conservation laws the asymptotic decay of periodic solutions of nonlinear hyperbolic conservation laws is an important nonlinear phenomenon it is observed that the genuine nonlinearity of equations forces the waves of each characteristic family to interact vigorously and to cancel each other the insightful analysis of glimm lax gl for scalar equations and 2 2 systems has indicated that the resultant mutual cancellation of interacting shock and rarefaction waves of the same family induces the decay of periodic solutions such a result was first shown by lax l1 in 1957 for one dimensional convex scalar conservation laws dafermos d1 applying his uniform processes proved the decay result for the case that the set of inflection points of the flux does not have an accumulation point the glimm lax theory gl indicates that for 2 2 strictly hyperbolic and genuinely nonlinear systems any periodic glimm solution decays like o 1 t this result was proved by using the approximate characteristic method in the glimm difference solutions provided that the oscillation of the corresponding initial data is small recently using the method of generalized characteristics dafermos d3 showed that any periodic solution with local bounded variation and small oscillation for 2 2 systems decays asymptotically with the same detailed structure pas found by lax l1 for the

conservation laws of mixed type describing three phase flow in porous media in this paper we examine the mathematical structure of a model for three phase incompressible flow in a porous medium we show that in the absence of diffusive forces the system of conservation laws describing the flow is not necessarily hyperbolic we present an example in which there is an elliptic region in saturation space for reasonable relative permeability data a linearized analysis shows that in nonhyperbolic regions solutions grow exponentially however the nonhyperbolic region if present will be of limited extent which inherently limits the exponential growth to examine these nonlinear effects we resort to fine grid numerical experiments with a suitably dissipative numerical method these experiments indicate that the solutions of riemann problems remain well behaved in spite of the presence of a linearly unstable elliptic region in saturation space in particular when initial states are outside the elliptic region the riemann problem solution appears to stay outside the region further

percolation in networks with long range connections abstract two dimensional lattices of points are connected with long range links whose lengths are distributed according to p r r by changing the decay exponent one can go from d dimensional short range networks to dimensional networks topologically similar to random graphs percolation on these networks is numerically studied for systems of up to 10 7 sites the shortest path fractal and chemical dimensions are determined at the critical threshold as a function of the decay exponent

an exact riemann solver for compressible two phase flow models containing non conservative products in this article we present a new numerical procedure for solving exactly the riemann problem of compressible two phase flow models containing non conservative products these products appear in the expressions for the interactions between the two phases thus in the compressible limit the governing equations are hyperbolic but can not be written as conservation laws i e in divergence form in general the solution to the riemann problem of these models contains six distinct centered waves according to the relative position of these waves in the x t plane the possible solutions can be classified into four principal configurations the riemann solver we propose herein investigates sequentially each of these configurations until an admissible solution is calculated special configurations corresponding to coalescence of waves are also analyzed and included in the solver further we examine the accuracy and robustness of three known methods for the integration of the non conservative products via a series of numerical tests finally the issue of existence and uniqueness of solutions to the riemann problem is discussed

a case study of global stability of strong rarefaction waves for 2 2 hyperbolic conservation laws with artificial viscosity this paper is concerned with the global stability of strong rarefaction waves for a class of 2 x 2 hyperbolic conservation laws with artificial viscosity i e the p system with artificial viscosity

bifurcation of nonclassical viscous shock profiles from the constant state we determine the bifurcation from the constant solution of nonclassical transitional and overcompressive viscous shock profiles in regions of strict hyperbolicity whereas classical shock waves in systems of conservation laws involve a single characteristic field nonclassical waves involve two fields in an essential way this feature is reflected in the viscous profile differential equation which undergoes codimension three bifurcation of the kind studied by dumortier et al as opposed to the codimension one bifurcation occurring in the classical case we carry out a complete bifurcation analysis for systems of two quadratic conservation laws with constant strictly parabolic viscosity matrices by reducing to a canonical form introduced by fiddelaers we show that all such systems except possibly those on a codimension one variety in parameter space give rise to nonclassical shock waves and we classify the number and types of their bifurcation points one consequence of our analysis is that weak transitional waves arise in pairs with profiles forming a 2 cycle configuration previously shown to lead to nonuniqueness of riemann solutions and to nontrivial asymptotic dynamics of the conservation laws another consequence is that appearance of weak nonclassical waves is necessarily associated with change of stability in constant solutions of the parabolic system of conservation laws rather than with change of type in the associated hyperbolic system