convergence to equilibrium for the relaxation approximations of conservation laws we study the cauchy problem for 2 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term special comparison and compactness properties are established by assuming the subcharacteristic condition therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0 this research was strongly motivated by the recent numerical investigations of s jin and z xin on the relaxation schemes for conservation laws 1996 john wiley sons inc

convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping we consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time asymptotically the hyperbolic model may be viewed as isentropic euler equations with friction term added to the momentum equation to model gas flow through a porous media in this case our result justifies darcy s law time asymptotically our model may also be viewed as an elastic model with damping

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

on relaxation hyperbolic systems violating the shizuta kawashima condition in this paper we start a general study on relaxation hyperbolic systems which violate the shizuta kawashima sk coupling condition this investigation is motivated by the fact that this condition is not satisfied by various physical systems and almost all the time in several space dimensions first we explore the role of entropy functionals around equilibrium solutions which may not be constant proposing a stability condition for such solutions then we find strictly dissipative entropy functions for one dimensional 2 2 systems which violate the sk condition finally we prove the existence of global smooth solutions for a class of systems such that condition sk does not hold but which are linearly degenerated in the non dissipative directions

stability and hopf bifurcation of the stationary solutions to an epidemic model with cross diffusion in the previous paper cai and wang 2015 we investigated the stationary solutions of a cross diffusion epidemic model with vertical transmission in a spatially heterogeneous environment with neumann boundary condition and proved that the set of positive stationary solutions forms a bounded bifurcation branch which is monotone s or fish hook shaped with respect to the bifurcation parameter in the present paper we give some criteria on the stability of solutions on we prove that the stability of the solutions changes only at every turning point of while in a different case that a k and x are sufficiently large original stable positive stationary solutions at certain point may lose their stability and hopf bifurcation can occur these results are very different from those of the spatially homogeneous or without cross diffusion cases

traveling waves in a nonlocal dispersal sirh model with relapse this paper is concerned with traveling wave solutions of a nonlocal dispersal susceptibleinfectiveremovalhealing for short sirh model with relapse it is found that the existence and nonexistence of traveling waves of the system are not only determined by the critical wave speed c but also by the basic reproduction number r0 of the corresponding system of ordinary differential equations more precisely we use schauders fixed point theorem to obtain the existence of traveling waves for r0 1 and c c and the nonexistence of traveling waves for r0 1 and 0

on hyperbolic dissipative systems of composite type abstract the shizuta kawashima condition plays the fundamental role in guaranteeing global stability for systems of hyperbolic parabolic hyperbolic with relaxation however there are many important physical systems not satisfying this coupling condition which are of composite type with regard to dissipation the compressible navier stokes equations with zero heat conductivity and euler equations of adiabatic flow through porous media are two typical examples in this paper we construct the global unique solution near constant equilibria to these two systems in three dimensions for the small h l l 3 initial data our proof is based on a reformation of the systems in terms of the pressure velocity and entropy a scaled energy estimates with minimal fractional derivative counts in conjunction with the linear l 2 l 2 decay estimates to extract a fast enough decay of velocity gradient which is used to close the energy estimates for the non dissipative entropy we also include an application to certain two phase models

zero relaxation limit to centered rarefaction waves for a rate type viscoelastic system abstract we study a rate type viscoelastic system proposed by i suliciu 1990 internat j engrg sci 28 827 841 which is a 3 3 hyperbolic system with relaxation as the relaxation time tends to zero this system convergences to the well known p system formally in the case where the initial data are the riemann data such that the corresponding solutions of the p system are centered rarefaction waves we show that if the wave strength is suitably small then the solution for the relaxation system exists globally in time and converges to the solution of the corresponding rarefaction waves uniformly as the relaxation time goes to zero except for an initial layer the jump discontinuities in the solutions are decaying exponentially fast as time tends to infinity

asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy we study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions under the shizuta kawashima condition we show that these solutions approach constant equilibrium state in the lp norm at a rate o t m 2 1 1 p as t tends to infty for p in min m 2 infty moreover we can show that we can approximate with a faster order of convergence theconservative part of the solution in terms of the linearized hyperbolic operator for m 2 and by a parabolic equation in the spirit of chapman enskog expansion the main tool is given by a detailed analysis of the green function for the linearized problem

gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation we study gas flow in vibrational nonequilibrium the model is a 4 4 nonlinear hyperbolic system with relaxation under physical assumptions properties of thermodynamic variables relevant to stability are obtained global existence for cauchy problems with smooth and small data is established and large time behavior is studied in the pointwise sense we formulate the fundamental solution in a systematic way for a general linear system with relaxation the fundamental solution provides insights to the behavior of the nonlinear system and is crucial to obtain our pointwise asymptotic picture for the nonequilibrium flow we also clarify in a general setting the relation between subcharacteristic conditions and a dissipative criterion that was originally proposed for hyperbolic parabolic systems and has now proved to be important also for hyperbolic systems with relaxation

preferential urn model and nongrowing complex networks a preferential urn model which is based on the concept the rich get richer is proposed from a relationship between a nongrowing model for complex networks and the preferential urn model in regard to degree distributions it is revealed that a fitness parameter in the nongrowing model is interpreted as an inverse local temperature in the preferential urn model furthermore it is clarified that the preferential urn model with randomness generates a fat tailed occupation distribution the concept of the local temperature enables us to understand the fat tailed occupation distribution intuitively since the preferential urn model is a simple stochastic model it can be applied to research on not only the nongrowing complex networks but also many other fields such as econophysics and social sciences

s shaped global bifurcation curve and hopf bifurcation of positive solutions to a predator prey model where d is a bounded domain in r n 1 with smooth boundary d and a b c d m are positive constants problem 1 0 is known as the predator prey model with holling tanner type interactions where u and v represent the densities of the prey and predator respectively hence we are only interested in positive solutions by which we mean that both u and v are positive in d it is easy to see that a 1 is necessary for the existence of positive solutions to 1 0 and we shall also assume that d 1 the case d 1 is rather different and will not be discussed here we are mainly interested in studying the number of positive solutions of 1 0 and the stability of these solutions in particular we shall show that when b d c and m fall into certain range the solution set u v a of 1 0 forms a s shaped smooth curve and that hopf bifurcation occurs along this curve this not only confirms rigorously similar numerical observations on 1 0 made in 4 but also shows that the corresponding parabolic system

condensation phenomena with distinguishable particles we study real space condensation phenomena in types of classical stochastic processes site particle system such as zero range processes and urn models we study a stochastic process in the ehrenfest class i e particles in a site are distinguishable in terms of the statistical mechanical analog the ehrenfest class obeys the maxwell boltzmann statistics we analytically clarify conditions for condensation phenomena in disordered cases in the ehrenfest class in addition we discuss the preferential urn model as an example of the disordered urn model it becomes clear that the quenched disorder property plays an important role in the occurrence of the condensation phenomenon in the preferential urn model it is revealed that the preferential urn model shows three types of condensation depending on the disorder parameters

analytic study of the three urn model for separation of sand we present an analytic study of the urn model for separation of sand recently introduced by lipowski and droz phys rev e 65 031307 2002 we solve analytically the master equation and the first passage problem the analytic results confirm the numerical results obtained by lipowski and droz we find that the stationary probability distribution and the shortest one among the characteristic times are governed by the same free energy we also analytically derive the form of the critical probability distribution on the critical line which supports their results obtained by numerically calculating binder cumulants a lipowski and m droz e print cond mat 0201472

condensation in perturbed zero range processes condensation phenomena in zero range processes have attracted significant attention recently here we consider zero range processes in which n sites are occupied by m particles that jump between sites according to a jump rate function given by g let z z1 z2 zn denote the steady state of a zero range process and let z n be the size of the maximum cluster a non complete condensation is defined as an event in which z n n converges to a positive constant less than 1 as n tends to infinity in this work we provide evidence that condensation depends on the detailed form of the jump rates by showing that non complete condensation occurs when g is given by where mk are bounded constants satisfying suitable conditions note that this choice of g is a perturbation of which exhibits perfect condensation i e n z n converges to 0 as n tends to infinity

construction of bgk models with a family of kinetic entropies for a given system of conservation laws we introduce a general framework for kinetic bgk models we assume to be given a system of hyperbolic conservation laws with a family of lax entropies and we characterize the bgk models that lead to this system in the hydrodynamic limit and that are compatible with the whole family of entropies this is obtained by a new characterization of maxwellians as entropy minimizers that can take into account the simultaneous minimization problems corresponding to the family of entropies we deduce a general procedure to construct such bgk models and we show how classical examples enter the framework we apply our theory to isentropic gas dynamics and full gas dynamics and in both cases we obtain new bgk models satisfying all entropy inequalities

yang lee zeros for an urn model for the separation of sand we apply the yang lee theory of phase transitions to an urn model for the separation of sand the effective partition function of this nonequilibrium system can be expressed as a polynomial of the size dependent effective fugacity z numerical calculations show that in the thermodynamic limit the zeros of the effective partition function are located on the unit circle in the complex z plane in the complex plane of the actual control parameter certain roots converge to the transition point of the model thus the yang lee theory can be applied to a wider class of nonequilibrium systems than those considered previously

hyperbolic limit of the jin xin relaxation model we consider the special jin xin relaxation model 0 1 u i a u u x u xx u tt we assume that the initial data u 0 u 0 t are sufficiently smooth and close to u 0 in l and have small total variation then we prove that there exists a solution u t u t t with uniformly small total variation for all t 0 and this solution depends lipschitz continuously in the l 1 norm with respect to time and the initial data letting 0 the solution u converges to a unique limit providing a relaxation limit solution to the quasi linear nonconservative system 0 2 u t a u u x 0 these limit solutions generate a lipschitz semigroup s on a domain d containing the functions with small total variation and close to u this is precisely the riemann semigroup determined by the unique riemann solver compatible with 0 1

discontinuous condensation transition and nonequivalence of ensembles in a zero range process we study a zero range process where the jump rates do not only depend on the local particle configuration but also on the size of the system rigorous results on the equivalence of ensembles are presented characterizing the occurrence of a condensation transition in contrast to previous results the phase transition is discontinuous and the system exhibits ergodicity breaking and metastable phases this leads to a richer phase diagram including nonequivalence of ensembles in certain phase regions the paper is motivated by results from granular clustering where these features have been observed experimentally

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

the principle of linearized stability for quasilinear parabolic evolution equations on donne des conditions pour la stabilite des points d equilibre des equations paraboliques quasilineaires avec des conditions aux limites non lineaires u a t u u u f t u 7b b t u u 0 u t 0 u 0 t 0 t

uniform global existence and parabolic limit for partially dissipative hyperbolic systems abstract this work concerns smooth solutions to the cauchy problem for first order partially dissipative hyperbolic systems with a small parameter the systems are written in non conservative form in several space variables we introduce algebraic conditions on the structure of the systems under these conditions together with a partial dissipation condition and the shizuta kawashima stability condition we prove three main results around constant equilibrium states these results are uniform global existence with respect to the parameter global in time convergence of the systems to second order nonlinear parabolic systems in a slow time variable and global existence when the parameter is fixed we also give examples of physical models to which the above results can be applied

large time asymptotics for partially dissipative hyperbolic systems this work is concerned with n component hyperbolic systems of balance laws in m space dimensions first we consider linear systems with constant coefficients and analyze the possible behavior of solutions as t using the fourier transform we examine the role that control theoretical tools such as the classical kalman rank condition play we build lyapunov functionals allowing us to establish explicit decay rates depending on the frequency variable in this way we extend the previous analysis by shizuta and kawashima under the so called algebraic condition sk in particular we show the existence of systems exhibiting more complex behavior than the one that the sk condition allows we also discuss links between this analysis and previous literature in the context of damped wave equations hypoellipticity and hypocoercivity to conclude we analyze the existence of global solutions around constant equilibria for nonlinear systems of balance laws our analysis of the linear case allows proving existence results in situations that the previously existing theory does not cover

a discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws abstract we present a new relaxation approximation to scalar conservation laws in several space variables by means of semilinear hyperbolic systems of equations with a finite number of velocities under a suitable multidimensional generalization of the whitham relaxation subcharacteristic condition we show the convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

multiple coexistence states for a prey predator system with cross diffusion abstract we study the multiple existence of positive solutions for the following strongly coupled elliptic system 1 v u u a u cv 0 in 1 u v v b du v 0 in u v 0 on where a b c d are positive constants and is a bounded domain in r n this is the steady state problem associated with a prey predator model with cross diffusion effects and u resp v denotes the population density of preys resp predators in particular the presence of represents the tendency of predators to move away from a large group of preys assuming that is small and that is large we show that the system admits a branch of positive solutions which is s or shaped with respect to a bifurcation parameter so that the system has two or three positive solutions for suitable range of parameters our method of analysis uses the idea developed by du lou j differential equations 144 1998 390 and is based on the bifurcation theory and the lyapunov schmidt procedure

global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy we consider the cauchy problem for a general one dimensional n n hyperbolic symmetrizable system of balance laws it is well known that in many physical examples for instance for the isentropic euler system with damping the dissipation due to the source term may prevent the shock formation at least for smooth and small initial data our main goal is to find a set of general and realistic sufficient conditions to guarantee the global existence of smooth solutions and possibly to investigate their asymptotic behavior for systems which are entropy dissipative a quite natural generalization of the kawashima condition for hyperbolic parabolic systems can be given in this paper we first propose a general framework for this kind of problem by using the so called entropy variables then we go on to prove some general statements about the global existence of smooth solutions under different sets of conditions in particular the present approach is suitable for dealing with most of the physical examples of systems with a relaxation extension our main tools will be some refined energy estimates and the use of a suitable version of the kawashima condition

dissipative structure and entropy for hyperbolic systems of balance laws in this paper we introduce an entropy condition for hyperbolic systems of balance laws under this condition we use the chapman enskog expansion to derive the corresponding viscous conservation laws further structural conditions are discussed in order to develop local and global existence theories for the balance laws and viscous conservation laws

discrete kinetic schemes for multidimensional systems of conservation laws we present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations which includes the relaxation schemes by s jin and z xin these schemes have a simple formulation even in the multidimensional case and do not need the solution of the local riemann problems for these approximations we give a suitable multidimensional generalization of the whitham s stability subcharacteristic condition in the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

the lp stability of relaxation rarefaction profiles we consider the large time behavior of solutions for a hyperbolic relaxation system for a certain class of initial data the solution is shown to converge to relaxation rarefaction profiles at a determined asymptotic rate the result is established without the smallness conditions of the wave strength and the initial disturbances 2001 academic press

semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation we investigate totally linearly degenerate hyperbolic systems with relaxation we aim to study their semilinear behavior which means that the local smooth solutions cannot develop shocks and the global existence is controlled by the supremum bound of the solution in this paper we study two specific examples the it suliciu type and the it kerr debye type models for the suliciu model which arises from the numerical approximation of isentropic flows the semilinear behavior is obtained using pointwise estimates of the gradient for the kerr debye systems which arise in nonlinear optics we show the semilinear behavior via energy methods for the original kerr debye model thanks to the special form of the interaction terms we can show the global existence of smooth solutions

stability of steady state solutions to a prey predator system with cross diffusion abstract this paper is concerned with a cross diffusion system arising in a prey predator population model the main purpose is to discuss the stability analysis for coexistence steady state solutions obtained by kuto and yamada j differential equations to appear we will give some criteria on the stability of these coexistence steady states furthermore we show that the hopf bifurcation phenomenon occurs on the steady state solution branch under some conditions

hyperbolic principal subsystems entropy convexity and subcharacteristic conditions we consider a system of n balance laws compatible with an entropy principle and convex entropy density using the special symmetric form induced by the main field we define the concept of principal subsystem associated with the system we prove that the 2 n 2 principal subsystems are also symmetric hyperbolic and satisfy a subentropy law moreover we can verify that for each principal subsystem the maximum minimum characteristic velocity is not larger smaller than the maximum minimum characteristic velocity of the full system these are the subcharacteristic conditions we present some simple examples in the case of the euler fluid then in the case of dissipative hyperbolic systems we consider an equilibrium principal subsystem and we discuss the consequences in the setting of extended thermodynamics finally in the moments approach to the boltzmann equation we prove as a consequence of the previous result that the maximum characteristic velocity evaluated at the equilibrium state does not decrease when the number of moments increases

analytic study of clustering in shaken granular material using zero range processes abstract we show that models used to describe granular clustering due to vertical shaking belong to the class of zero range processes this correspondence allows us to derive analytically in the large particle number limit in a very easy and straightforward manner a number of properties of the models like particle distribution functions phase diagram and characteristic time of clusterization

exact representation of the asymptotic drift speed and diffusion matrix for a class of velocity jump processes abstract this paper examines a class of linear hyperbolic systems which generalizes the goldstein kac model to an arbitrary finite number of speeds v i with transition rates i j under the basic assumptions that the transition matrix is symmetric and irreducible and the differences v i v j generate all the space the system exhibits a large time behavior described by a parabolic advection diffusion equation the main contribution is to determine explicit formulas for the asymptotic drift speed and diffusion matrix in term of the kinetic parameters v i and i j establishing a complete connection between microscopic and macroscopic coefficients it is shown that the drift speed is the arithmetic mean of the velocities v i the diffusion matrix has a more complicate representation based on the graph with vertices the velocities v i and arcs weighted by the transition rates i j the approach is based on an exhaustive analysis of the dispersion relation and on the application of a variant of the kirchoff s matrix tree theorem from graph theory

global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow abstract we establish global solutions of nonconcave hyperbolic equations with relaxation arising from traffic flow one of the characteristic fields of the system is neither linearly degenerate nor genuinely nonlinear furthermore there is no dissipative mechanism in the relaxation system characteristics travel no faster than traffic the global existence and uniqueness of the solution to the cauchy problem are established by means of a finite difference approximation to deal with the nonconcavity we use a modified argument of oleinik amer math soc translations 26 1963 95 it is also shown that the zero relaxation limit of the solutions exists and is the unique entropy solution of the equilibrium equation

coexistence of a cross diffusive dengue fever model in a heterogeneous environment abstract a spatio temporal sis si dengue model with cross diffusion is formulated as the movements of human and mosquitoes have been intricately linked with the spread of dengue fever to highlight the impacts of cross diffusion on the dynamical processes we focus on the nonnegative steady state solutions of the dengue model that is the coexistence of the corresponding strongly coupled elliptic system by means of the relevant eigenvalue problem we investigate some properties of the basic reproduction number to the model and further present the existence of coexistence solutions our results imply that the virus carried in human and that in mosquitoes can coexist if the basic reproduction number is greater than one and the extent of cross diffusion is small enough the final numerical simulations and epidemiological explanations make our analytical findings easier to understand

global existence for small data of the viscous green naghdi type equations abstract we consider the cauchy problem for the green naghdi equations with viscosity for small initial data it is well known that adding a second order dissipative term to a hyperbolic system leads to the existence of global smooth solutions once the hyperbolic system is symmetrizable and the so called kawashima shizuta condition is satisfied in a previous work we have proved that the green naghdi equations can be written in a symmetric form using the associated hamiltonian this system being dispersive in the sense that it involves third order derivatives the symmetric form is based on symmetric differential operators in this paper we use this structure for an appropriate change of variable to prove that adding viscosity effects through a second order term leads to global existence of smooth solutions for small data we also deduce that constant solutions are asymptotically stable

entropy and global existence for hyperbolic balance laws this paper presents a general result on the existence of global smooth solutions to hyperbolic systems of balance laws in several space variables we propose an entropy dissipation condition and prove the existence of global smooth solutions under initial data close to a constant equilibrium state in addition we show that a system of balance laws satisfies a kawashima condition if and only if its first order approximation namely the hyperbolic parabolic system derived through the chapman enskog expansion satisfies the corresponding kawashima condition the result is then applied to bouchut s discrete velocity bgk models approximating hyperbolic systems of conservation laws

large time behavior of solutions for hyperbolic balance laws we study the existence and the large time behavior of global solutions to the initial value problem for hyperbolic balance laws in n space dimensions with n 3 admitting an entropy and satisfying the stable condition we first construct global existence of the solutions to such a system around a steady state if the initial energy is small enough then we show that k order derivatives of these solutions approach a constant state in the lp norm at a rate o t 12 k n2 np with p 2 and 0 n2 provided that initially z0 b 2 where b 2 is a homogeneous besov space these decay results do not impose an additional smallness assumption on lp norm of the initial data and we thus improve the results in 3 19 we also show faster decay results in the sense that if pz0 b 2 i p z0 b 2 1 with n2 n 22 k order derivatives of the solutions approach a constant state in the lp norm at a rate o t 12 k 1 n2 np

positive steady states in an epidemic model with nonlinear incidence rate abstract in this paper we investigate a diffusive epidemic model with nonlinear incidence rate under homogeneous neumann boundary condition the value of this study lies in two aspects mathematically we show the stability of the constant positive steady state solution the existence and nonexistence the local and global structure of nonconstant positive steady states and epidemiologically we find that the system exhibits turing patterns controlled by diffusion

materials with internal variables and relaxation to conservation laws the theory of materials with internal state variables of coleman gurtin cg provides a natural framework to investigate the structure of relaxation approximations of conservation laws from the viewpoint of continuum thermomechanics after reviewing the requirements imposed on constitutive theories by the principle of consistency with the clausius duhem inequality we pursue two specific theories pertaining to stress relaxation and relaxation of internal energy they each lead to a relaxation framework for the theory of thermoelastic non conductors of heat equipped with globally defined entropy functions for the associated relaxation process next we consider a semilinear model problem of stress relaxation we discuss uniform stability and compactness for solutions of the relaxation system in the zero relaxation limit and establish convergence to the system of isothermal elastodynamics by using compensated compactness finally we prove a strong dissipation estimate for the relaxation approximations proposed in jin xin jx when the limit system is equipped with a strictly convex entropy

urn model of separation of sand we introduce an urn model that describes spatial separation of sand in this dynamical model in a certain range of parameters spontaneous symmetry breaking takes place and equipartitioning of sand into two compartments is broken the steady state equation for an order parameter a critical line and the tricritical point on the phase diagram are found exactly the master equation and the first passage problem for the model are solved numerically and the results are used to locate first order transitions exponential divergence of a certain characteristic time shows that the model can also exhibit very strong metastability in certain cases characteristic time diverges as n z where n is the number of balls and z frac 1 2 critical line frac 2 3 tricritical point or frac 1 3 limits of stability