linear stability of propagating phase boundaries in capillary fluids abstract dynamical phase changes occurring along planar traveling waves in capillary fluids are considered their multi dimensional stability which is encoded in the spectrum of a third order variational differential operator is addressed by two approaches on one hand an original energy method is used to show that these diffuse interfaces are weakly stable more precisely the spectrum of the involved operator is found to coincide with the imaginary axis on the other hand an evans function technique reveals a bifurcation phenomenon about zero the interpretation of which still needs some clarification it is the counterpart of the surface waves exhibited in an earlier work for the corresponding sharp interfaces

asymptotic stability of traveling wave solutions for perturbations with algebraic decay abstract for a class of scalar partial differential equations that incorporate convection diffusion and possibly dispersion in one space and one time dimension the stability of traveling wave solutions is investigated if the initial perturbation of the traveling wave profile decays at an algebraic rate then the solution is shown to converge to a shifted wave profile at a corresponding temporal algebraic rate and optimal intermediate results that combine temporal and spatial decay are obtained the proofs are based on a general interpolation principle which says that algebraic decay results of this form always follow if exponential temporal decay holds for perturbation with exponential spatial decay and the wave profile is stable for general perturbations

decay rates to viscous contact waves for the compressible navier stokes equations in this paper we study the large time asymptotic behavior of contact wave for the cauchy problem of one dimensional compressible navier stokes equations with zero viscosity when the riemann problem for the euler system admits a contact discontinuity solution we can construct a contact wave which approximates the contact discontinuity on any finite time interval for small heat conduction and then runs away from it for large time and proves that it is nonlinearly stable provided that the strength of the contact discontinuity and the perturbation of the initial data are suitably small

stability of viscous shock waves associated with a system of nonstrictly hyperbolic conservations laws we introduce a simple model of two conservation laws which is strictly hyperbolic except for a degenerate parabolic line in the state space besides classical shock waves it also exhibits overcompressive marginal overcompressive and marginal undercompressive shock waves our purpose is to study the behavior of the corresponding viscous waves in particular the manner in which these waves are stable there are several basic differences between classical shock waves and other types of shock waves a perturbation of an overcompressive shock wave gives rise to a new wave monotone marginal overcompressive waves behave distinctly from the nonmonotone ones analytical techniques used in our study include characteristic energy and weighted energy methods and nonlinear superposition through time invariants although we carry out our analysis for a simple model the general phenomena would hold for overcompressive waves which occur in other physical models

rank driven markov processes we study a class of markovian systems of n elements taking values in 0 1 that evolve in discrete time t via randomized replacement rules based on the ranks of the elements these rank driven processes are inspired by variants of the bak sneppen model of evolution in which the system represents an evolutionary fitness landscape and which is famous as a simple model displaying self organized criticality our main results are concerned with long time large n asymptotics for the general model in which at each time step k randomly chosen elements are discarded and replaced by independent u 0 1 variables where the ranks of the elements to be replaced are chosen independently at each time step according to a distribution n on 1 2 n k our main results are that under appropriate conditions on n the system exhibits threshold behavior at s 0 1 where s is a function of n and the marginal distribution of a randomly selected element converges to u s 1 as t and n of this class of models results in the literature have previously been given for special cases only namely the mean field or random neighbor bak sneppen model our proofs avoid the heuristic arguments of some of the previous work and use foster lyapunov ideas our results extend existing results and establish their natural more general context we derive some more specialized results for the particular case where k 2 one of our technical tools is a result on convergence of stationary distributions for families of uniformly ergodic markov chains on increasing state spaces which may be of independent interest

asymptotic stability of rarefaction wave for the navier stokes equations for a compressible fluid in the half space this paper is concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the navier stokes equations in a compressible fluid in the eulerian coordinate in the half space this is the second one of our series of papers on this subject in this paper firstly we classify completely the time asymptotic states according to some parameters that is the spatial asymptotic states and boundary conditions for this initial boundary value problem and some pictures for the classification of time asymptotic states are drawn in the state space in order to prove the stability of the rarefaction wave we use the solution to burgers equation to construct a suitably smooth approximation of the rarefaction wave and establish some time decay estimates in lp norm for the smoothed rarefaction wave we then employ the l2 energy method to prove that the rarefaction wave is non linearly stable under a small perturbation as time goes to infinity

characteristic boundary layers in real vanishing viscosity limits abstract in this paper we study the vanishing viscosity limit of initial boundary value problems for one dimensional mixed hyperbolic parabolic systems when the boundary is characteristic for both the viscous and the inviscid systems in particular we assume that an eigenvalue of the inviscid system vanishes uniformly we prove the stability of boundary layers expansions in small time i e before shocks for the inviscid system as long as the amplitude of the boundary layers remains sufficiently small in particular by using lagrangian coordinates we apply our result to physical systems like gasdynamics and magnetohydrodynamics with homogeneous dirichlet condition for the velocity at the boundary

multidimensional viscous shocks ii the small viscosity limit in this paper we prove the existence of curved multidimensional viscous shocks and also justify the small viscosity limit starting with a curved multidimensional inviscid shock solution to a system of hyperbolic conservation laws we show that the shock can be obtained as a small viscosity limit of solutions to an associated parabolic problem viscous shocks the two main hypotheses are a natural evans function assumption on the viscous profile together with a restriction on how much the shock can deviate from flatness the main tools are a conjugation lemma that removes xn dependence from the linearization of the parabolic problem about the viscous profile new degenerate kreiss type symmetrizers used to prove an l2 estimate for the linearized problem and a finite regularity calculus of semiclassical and mixed type classical semiclassical pseudodifferential operators 2003 wiley periodicals inc

l2 contraction of large planar shock waves for multi dimensional scalar viscous conservation laws abstract we consider a l 2 contraction a l 2 type stability of large viscous shock waves for the multi dimensional scalar viscous conservation laws up to a suitable shift by using the relative entropy methods quite different from the previous results we find a new way to determine the shift function which depends both on the time and space variables and solves a viscous hamilton jacobi type equation with source terms moreover we do not impose any conditions on the anti derivative variables of the perturbation around the shock profile more precisely it is proved that if the initial perturbation around the viscous shock wave is suitably small in l 2 norm then the l 2 contraction holds true for the viscous shock wave up to a suitable shift function note that bv norm or the l norm of the initial perturbation and the shock wave strength can be arbitrarily large furthermore as the time t tends to infinity the l 2 contraction holds true up to a spatially homogeneous time dependent shift function in particular if we choose some special initial perturbations then l 2 convergence of the solutions towards the associated shock profile can be proved up to a time dependent shift

hydrodynamic limit of the boltzmann equation with contact discontinuities the hydrodynamic limit for the boltzmann equation is studied in the case when the limit system that is the system of euler equations contains contact discontinuities when suitable initial data is chosen to avoid the initial layer we prove that there exist a family of solutions to the boltzmann equation globally in time for small knudsen number and this family of solutions converge to the local maxwellian defined by the contact discontinuity of the euler equations uniformly away from the discontinuity as the knudsen number e tends to zero the proof is obtained by an appropriately chosen scaling and the energy method through the micro macro decomposition

construction of confidence limits about effect measures a general approach it is widely accepted that confidence interval construction has important advantages over significance testing for the presentation of research results as now facilitated by readily available software however for a number of effect measures procedures are either not available or not satisfactory in samples of small to moderate size in this paper we describe a general approach for estimating a difference between effect measures which can also be used to obtain confidence limits for a risk ratio and a lognormal mean numerical evaluation shows that this closed form procedure outperforms existing methods including the bootstrap

decay rate for perturbations of stationary discrete shocks for convex scalar conservation laws this paper is to study the decay rate for perturbations of stationary discrete shocks for the lax friedrichs scheme approximating the scalar conservation laws by means of an elementary weighted energy method if the summation of the initial perturbation over j is small and decays at the algebraic rate as j then the solution approaches the stationary discrete shock profiles at the corresponding rate as n this rate seems to be almost optimal compared with the rate in the continuous counterpart proofs are given by applying the weighted energy integration method to the integrated scheme of the original one the selection of the time dependent discrete weight function plays a crucial role in this procedure

ion acoustic shock in a collisional plasma abstract the paper is concerned with the propagation of ion acoustic shock waves in a collision dominated plasma whose equations of motion are described by the one dimensional isothermal navier stokes poisson system for ions with the electron density determined by the boltzmann relation the main results include three parts a we establish the existence and uniqueness of a small amplitude smooth traveling wave by solving a 3 d ode in terms of the center manifold theorem b we study the shock structure in a specific asymptotic regime where the viscosity coefficient and the shock strength are proportional to e and the debye length is proportional to e 1 2 with two parameters e and and show that in the limit e 0 shock profiles obtained in a can be approximated by the profiles of kdv burgers uniformly for 0 0 with some 0 0 the proof is based on the suitable construction of the kdv burgers shock profiles together with the delicate analysis of a linearized variable coefficient system in exponentially weighted sobolev spaces involving parameters e and c we also prove the large time asymptotic stability of traveling waves under suitably small smooth zero mass perturbations note that the ions temperature is allowed to be zero in parts a and b but necessarily required to be strictly positive in the proof of part c

asymptotic fitness distribution in the bak sneppen model of biological evolution with four species we suggest a new method to compute the asymptotic fitness distribution in the bak sneppen model of biological evolution as applications we derive the full asymptotic distribution in the four species model and give an explicit linear recurrence relation for a set of coefficients determining the asymptotic distribution in the five species model

nonlinear stability of shock fronts for a relaxation system in several space dimensions it is proposed by jin and xin in 3 to use the following relaxation system u t sum limits i 1 m left upsilon i right x i 0 text u in r n text upsilon i in r n 1 upsilon it a i u x i frac left upsilon i f i left u right right varepsilon text i 1 2 ldots m 2 to approximate multi dimensional hyperbolic conservation laws of the form u t sum limits i 1 m f i left u right x i 0 left x i t right in r 1 times r u in r n 3 here e is a positive constant representing the rate of relaxation and a i a i i a i 0 i 1 2 m with i being the n n identity matrix and a i being positive constants satisfying the subcharacteristic condition see 3 and sc one of the main advantages of the system 1 2 is its form of local relaxation structure and linearity in convection which makes it possible to solve this system quite easily by underresolved stable numerical discretizations using neither riemann solvers spatially nor nonlinear system of algebraic equations solvers temporally a class of numerical schemes solving 1 2 has been proposed in 3 which yields satisfactory numerical solutions to hyperbolic conservation laws 3 3 yet the asymptotic equivalence of the relaxation system 1 2 and the hyperbolic conservation laws 3 as either the rate of relaxation goes to zero or time approaches infinity has been rigorously analysed only in one space dimension see 5 9 and 11

stability of contact discontinuity for jin xin relaxation system abstract in this paper we first construct a contact wave for 1 dimensional jin xin relaxation system s jin z p xin the relaxation schemes for systems of conservation laws in arbitrary space dimensions comm pure appl math 48 1995 235 277 this wave serves as the relaxation version of contact discontinuity of the corresponding hyperbolic system at equilibrium such a contact wave is shown to be nonlinearly stable under small initial perturbation the time decay rate is also obtained by weighted energy estimates

stability of rarefaction waves of the navier stokes poisson system abstract in the paper we are concerned with the large time behavior of solutions to the one dimensional navier stokes poisson system in the case when the potential function of the self consistent electric field may take distinct constant states at x precisely it is shown that if initial data are close to a constant state with asymptotic values at far fields chosen such that the riemann problem on the corresponding quasineutral euler system admits a rarefaction wave whose strength is not necessarily small then the solution exists for all time and tends to the rarefaction wave as t the construction of the nontrivial large time profile of the potential basing on the quasineutral assumption plays a key role in the stability analysis the proof is based on the energy method by taking into account the effect of the self consistent electric field on the viscous compressible fluid

local tracking and stability for degenerate viscous shock waves abstract we study the pointwise behavior of perturbed degenerate sonic shock waves for scalar conservation laws with nonconstant diffusion building on the pointwise green s function approach of zumbrun and howard indiana u math j 47 3 1998 741 we extend the linear analysis to an equation with nonintegrable coefficients in lieu of working with the integrated equation we employ a tracking mechanism that we expect will allow degenerate waves to be incorporated into the general framework for nondegenerate systems indiana u math j 47 3 1998 741

spectral stability of periodic wave trains of the korteweg de vries kuramoto sivashinsky equation in the korteweg de vries limit we study the spectral stability of a family of periodic wave trains of the korteweg de vries kuramoto sivashinsky equation tv v xv 3v 2 xv 4 xv 0 0 in the korteweg de vries limit 0 a canonical limit describing small amplitude weakly unstable thin lm ow more precisely we carry out a rigorous singular perturbation analysis reducing the problem of spectral stability in this limit to the validation of a set of three conditions each of which have been numerically analyzed in previous studies and shown to hold simultaneously on a non empty set of parameter space the main technical diculty

stability of viscous scalar shock fronts in several dimensions we prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions the proof uses the integrated equation and an effective equation for the motion of the front itself we derive energy estimates that balance terms from the integrated equation with terms from the front motion equation in this paper we prove that viscous shock profiles for scalar conservation laws are stable in two or more space dimensions these multidimensional stability questions are separate from their one dimensional analogues because of the possibility of transverse instabilities such as those that occur in combustion fronts lu and in shock waves with phase changes the proof here is a rigorous version of arguments that are used to derive effective equations such as the kuramoto sivashinsky equation to describe the behavior of fronts the one dimensional stability for scalar conservation laws was proven by ii in and oleinik io using the integrated equation see below and a max imum principle another proof based on weighted norms and spectral theory for the linearized problem was given by sattinger s the multidimensional stability proof below has more in common with the stability proofs for systems of conservation laws in one space dimension begun by kawashima and mat sumura km and goodman go and completed by liu li these proofs use l2 energy estimates for the integrated equation we consider equations of the form ut f u x g u y uxx uyy where f u is a strictly convex function of u

the validity of the modified equation for nonlinear shock waves abstract the modified model equivalent equation is an important tool in designing and analyzing nonlinear difference schemes in this note the validity of this principle is rigorously established for nonlinear shock wave solutions and the upwind scheme in a particular case

nonlinear stability of rarefaction waves for the boltzmann equation it is well known that the boltzmann equation is related to the euler and navier stokes equations in the field of gas dynamics the relation is either for small knudsen number or for dissipative waves in the time asymptotic sense in this paper we show that rarefaction waves for the boltzmann equation are time asymptotic stable and tend to the rarefaction waves for the euler and navier stokes equations our main tool is the combination of techniques for viscous conservation laws and the energy method based on micro macro decomposition of the boltzmann equation the expansion nature of the rarefaction waves and the suitable microscopic version of the h theorem are essential elements of our analysis

asymptotic stability of nonlinear wave for the compressible navier stokes equations in the half space in the present paper we investigate the large time behavior of the solution to an initial boundary value problem for the isentropic compressible navier stokes equations in the eulerian coordinate in the half space this is one of the series of papers by the authors on the stability of nonlinear waves for the outflow problem of the compressible navier stokes equations some suitable assumptions are made to guarantee that the time asymptotic state is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave employing the l2 energy method and making use of the techniques from the paper s kawashima y nikkuni stability of rarefaction waves for the discrete boltzmann equations adv math sci appl 12 1 2002 327 353 we prove that this nonlinear wave is nonlinearly stable under a small perturbation the complexity of nonlinear wave leads to many complicated terms in the course of establishing the a priori estimates however those terms are of two basic types and the terms of each type are good and can be evaluated suitably by using the decay in both time and space variables estimates of each component of nonlinear wave

stability of traveling waves for hamilton jacobi equations with finite speed perturbations abstract we prove nonlinear stability of planar shock for general hamilton jacobi equations with finite speed perturbation here we use energy estimates it is shown that the solution connecting a weak shock is asymptotically stable under small perturbations

stability of contact discontinuity for the boltzmann equation the boltzmann equation which describes the time evolution of a large number of particles through the binary collision in statistics physics has close relation to the systems of fluid dynamics that is euler equations and navier stokes equations as for a basic wave pattern to euler equations we consider the nonlinear stability of contact discontinuities to the boltzmann equation even though the stability of the other two nonlinear waves i e shocks and rarefaction waves has been extensively studied there are few stability results on the contact discontinuity because unlike shock waves and rarefaction waves its derivative has no definite sign and decays slower than a rarefaction wave moreover it behaves like a linear wave in a nonlinear setting so that its coupling with other nonlinear waves reveals a complicated interaction mechanism based on the new definition of contact waves to the boltzmann equation corresponding to the contact discontinuities for the euler equations we succeed in obtaining the time asymptotic stability of this wave pattern with a convergence rate in our analysis an intrinsic dissipative mechanism associated with this profile is found and used for closing the energy estimates

on a local version of the bak sneppen model a major difficulty in studying the bak sneppen model is in effectively comparing it with well understood models this stems from the use of two geometries complete graph geometry to locate the global fitness minimizer and graph geometry to replace the species in the neighborhood of the minimizer over the years a number of models inspired by bak sneppen were studied usually by introducing different or new features e g discretizing fitness randomized neighbors or population size we present a variant that only uses features present in bak sneppen and whose difference from the bak sneppen is that only the graph geometry is used for the evolution this allows to obtain the stationary distribution through random walk dynamics while preserving the geometric nature of the model we use this to show that for constant degree graphs the stationary fitness distribution converges to an iid law as the number of vertices tends to infinity we also discuss exponential ergodicity through coupling and avalanches for the model

a practical approach for comparing means of two groups without equal variance assumption in this paper we consider two groups of i i d normally distributed random variables n mu x sigma x 2 and n mu y sigma y 2 without assuming equal variance sigma x 2 sigma y 2 we propose a simple method for constructing confidence bounds based on howe s approximation i its applications in parallel clinical trial testing h 0 mu x mu y 0 versus h 1 mu x mu y 0 and parallel bioequivalence be trial testing h 0 mid r mu x mu y mid r delta versus h 1 mid r mu x mu y mid r delta are studied sample size calculation formulae for both cases are derived their performances are evaluated by simulation our study shows that the proposed procedure can control type i error satisfactorily compared with cochran cox s and satterthwaite s approximations while maintaining a relatively high power the proposed approach is not only simple for constructing the confidence limit but also provides a simple and accurate formula for sample size calculation

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

convergence to travelling fronts of solutions of the p system with viscosity in the presence of a boundary we study the asymptotic behavior as time goes to infinity of solutions to the initial boundary value problem on the half space r for a one dimensional model system for the isentropic flow of a compressible viscous gas the so called p system with viscosity as boundary conditions we prescribe the constant state at infinity and require that the velocity be zero at the boundary x 0 when the velocity at infinity is negative and satisfies a condition on the magnitude we prove that if the initial data are suitably close to those for the corresponding outgoing viscous shock profile which is suitably far from the boundary then a unique solution exists globally in time and tends toward the properly shifted viscous shock profile as the time goes to infinity the proof is given by an elementary energy method

asymptotic stability of relaxation shock profiles for hyperbolic conservation laws abstract this paper studies the asymptotic stability of traveling relaxation shock profiles for hyperbolic systems of conservation laws under a stability condition of subcharacteristic type the large time relaxation dynamics on the level of shocks is shown to be determined by the equilibrium conservation laws the proof is due to the energy principle using the weighted norms the interaction of waves from various modes is treated by imposing suitable weight matrix

limit behavior of the bak sneppen evolution model one of the key problems related to the bak sneppen evolution model on the circle is computing the limit distribution of the fitness at a fixed observation vertex in the stationary regime as the size of the system tends to infinity some simulations have suggested that this limit distribution is uniform on f 1 for some f 2 3 in this article we prove that the mean of the fitness in the stationary regime is bounded away from 1 uniformly in the size of the system thereby establishing the nontriviality of the limit behavior the bak sneppen dynamics can easily be defined on any finite connected graph we also present a generalization of the phase transition result in the context of an increasing sequence of such graphs this generalization covers the multidimentional bak sneppen model as well as the bak sneppen model on a tree our proofs are based on a self similar graphical representation of the avalanches

viscous limits for piecewise smooth solutions to systems of conservation laws in this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity it is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero the proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles

linear stability of shock profiles for systems of conservation laws with semi linear relaxation the evans function theory which has recently been applied to the study of linear stability of viscous shock profiles is developed below for semi linear relaxation we study the linear stability of shock profiles in the lax undercompressive and overcompressive cases the results we obtain are similar to those found for viscous approximations by gardner and zumbrun commun pure appl math 51 7 1998 797

on rank driven dynamical systems we investigate a class of models related to the bak sneppen bs model initially proposed to study evolution the bs model is extremely simple and yet captures some forms of complex behavior such as self organized criticality that is often observed in physical and biological systems in this model random fitnesses in 0 1 are associated to agents located at the vertices of a graph g their fitnesses are ranked from worst 0 to best 1 at every time step the agent with the worst fitness and some others with a priori given rank probabilities are replaced by new agents with random fitnesses we consider two cases the exogenous case where the new fitnesses are taken from an a priori fixed distribution and the endogenous case where the new fitnesses are taken from the current distribution as it evolves we approximate the dynamics by making a simplifying independence assumption we use order statistics and dynamical systems to define a rank driven dynamical system that approximates the evolution of the distribution of the fitnesses in these rank driven models as well as in the bs model for this simplified model we can find the limiting marginal distribution as a function of the initial conditions agreement with experimental results of the bs model is excellent

nonlinear stability of discrete shocks for systems of conservation laws in this paper we study the asymptotic nonlinear stability of discrete shocks for the lax friedrichs scheme for approximating general m systems of nonlinear hyperbolic conservation laws it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in the lp norm for all p 1 provided that the sums of the initial perturbations equal zero these results should shed light on the convergence of the numerical solution constructed by the lax friedrichs scheme for the single shock solution of system of hyperbolic conservation laws if the riemann solution corresponding to the given far field states is a superposition of m single shocks from each characteristic family we show that the corresponding multiple discrete shocks are nonlinearly stable in l p p 2 these results are proved by using both a weighted estimate and a characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the lax friedrichs scheme

stability of viscous shock waves associated with non convex modes in this paper we consider small amplitude viscous shock waves associated with a possibly non convex mode i e a not necessarily genuinely nonlinear field and prove their time asymptotic stability towards sufficiently small perturbations of not necessarily vanishing mass we thus extend the result of sx to the case of non convex modes and the result of f1 to the case of non zero mass perturbations

spectral stability of small shock waves this paper is the first in a series of articles that study the eigenvalue problem for small amplitude shock waves in systems of conservation laws with viscosity or relaxation the papers show in various contexts that in the zero amplitude limit appropriately scaled versions of the associated evans bundles converge to suspensions of evans bundles for fixed shock waves in related reduced systems with lower dimensional state space in this article the new geometric framework establishing this connection is introduced in the simplest context that of shock waves associated with a simple genuinely nonlinear mode in systems with identity viscosity in one space dimension

stability of a composite wave of two viscous shock waves for the full compressible navier stokes equation in this paper we investigate the asymptotic stability of a composite wave consisting of two viscous shock waves for the full compressible navier stokes equation by introducing a new linear diffusion wave special to this case we successfully prove that if the strengths of the viscous shock waves are suitably small with same order and also the initial perturbations which are not necessarily of zero integral are suitably small the unique global solution in time to the full compressible navier stokes equation exists and asymptotically tends toward the corresponding composite wave whose shifts in space of two viscous shock waves are uniquely determined by the initial perturbations we then apply the idea to study a half space problem for the full compressible navier stokes equation and obtain a similar result

scalar conservation laws with discontinuous flux function ii on the stability of the viscous profiles the equation frac partial u partial t frac partial partial x left h x f u left 1 h x right g u right 0 whereh is heaviside s step function appears for example in continuous sedimentation of solid particles in a liquid in two phase flow in traffic flow analysis and in ion etching the discontinuity of the flux function atx 0 causes a discontinuity of a solution which is not uniquely determined by the initial data by a viscous profile of this discontinuity we mean a stationary solution ofu t f x eu xx wheref is a smooth approximation of the discontinuous flux i e h is smoothed we present some results on the stability of the viscous profiles which means that a small disturbance tends to zero uniformly ast this is done by weighted energy methods where the weight depending onf andg plays a crucial role

vanishing viscosity limit of the compressible navier stokes equations for solutions to a riemann problem we study the vanishing viscosity limit of the compressible navier stokes equations to the riemann solution of the euler equations that consists of the superposition of a shock wave and a rarefaction wave in particular it is shown that there exists a family of smooth solutions to the compressible navier stokes equations that converges to the riemann solution away from the initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients this gives the first mathematical justification of this limit for the navier stokes equations to the riemann solution that contains these two typical nonlinear hyperbolic waves

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

on the thresholds probability densities and critical exponents of bak sneppen like models abstract we report a simple method to accurately determine the threshold and the exponent of the bak sneppen bs model and also investigate the bs universality class for the random neighbor version of the bs model we find the threshold x 0 33332 3 in agreement with the exact result x 1 3 given by mean field theory for the one dimensional original model we find x 0 6672 2 in good agreement with the results reported in the literature for the anisotropic bs model we obtain x 0 7240 1 we study the finite size effect x l x l l observed in a system with l sites and find 1 00 1 for the random neighbor version 1 40 1 for the original model and 1 58 1 for the anisotropic case finally we discuss the effect of defining the extremal site as the one which minimizes a general function f x instead of simply f x x as in the original updating rule we emphasize that models with extremal dynamics have singular stationary probability distributions p x our simulations indicate the existence of two symmetry based universality classes

stability of small amplitude boundary layers for mixed hyperbolic parabolic systems we consider an initial boundary value problem for a symmetrizable mixed hyperbolic parabolic system of conservation laws with a small viscosity e u e t f u e x e b u e u e x x when the boundary is noncharacteristic for both the viscous and the inviscid system and the boundary condition dissipative we show that u e converges to a solution of the inviscid system before the formation of shocks if the amplitude of the boundary layer is sufficiently small this generalizes previous results obtained for b invertible and the linear study of serre and zumbrun obtained for a pure dirichlet s boundary condition

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

pointwise estimates on the green s function for a scalar linear convection diffusion equation abstract pointwise estimates are found on green s functions for the scalar linear convection diffusion equations that arise when a scalar conservation law with non constant diffusion is linearized about a viscous shock profile of arbitrary strength the estimates take the form of gaussian kernels centered around paths determined by the typically different asymptotic states of the convection function the analysis extends the spectral transform method to the non constant coefficient case

criticality in simple models of evolution we consider two apparently similar models of biological evolution that have been claimed to exhibit self organized critical behavior a careful reanalysis of these models including several analytic results for one of them suggests that they are qualitatively different we demonstrate the limitations of the mean field description of these systems we argue that a more precise definition of self organized criticality is desirable and establish several criteria in this connection

stability of isentropic navier stokes shocks in the high mach number limit by a combination of asymptotic ode estimates and numerical evans function calculations we establish stability of viscous shock solutions of the isentropic compressible navier stokes equations with law pressure i in the limit as mach number m goes to infinity for any 1 proved analytically and ii for m 2 500 gamma in 1 2 5 or m 13 000 gamma in 2 5 3 demonstrated numerically this builds on and completes earlier studies by matsumura nishihara and barker humpherys rudd zumbrun establishing stability for low and intermediate mach numbers respectively indicating unconditional stability independent of shock amplitude of viscous shock waves for law gas dynamics in the range gamma in 1 3 other values may be treated similarly but have not been checked numerically the main idea is to establish convergence of the evans function in the high mach number limit to that of a pressureless or infinitely compressible gas with additional upstream boundary condition determined by a boundary layer analysis recall that low mach number behavior is formally incompressible

nonlinear stability of rarefaction waves for the compressible navier stokes equations with zero heat conductivity abstract this paper is concerned with the time asymptotic nonlinear stability of rarefaction waves to the cauchy problem of the one dimensional compressible navier stokes equations with zero heat conductivity under the assumption that the unique global entropy solution to the resulting riemann problem of the corresponding compressible euler equations consists of rarefaction waves only then if both the initial perturbation and the strengths of rarefaction waves are assumed to be suitably small we show that its cauchy problem admits a unique global solution which tends time asymptotically toward the rarefaction waves this result is proved by using the elementary energy method and the argument developed by kawashima and okada 1982

critical thresholds and the limit distribution in the bak sneppen model one of the key problems related to the bak sneppen evolution model is to compute the limit distribution of the fitnesses in the stationary regime as the size of the system tends to infinity simulations in 3 1 4 suggest that the one dimensional limit marginal distribution is uniform on p c 1 for some p c 0 667 in this paper we define three critical thresholds related to avalanche characteristics we prove that if these critical thresholds are the same and equal to some p c we can only prove that two of them are the same then the limit distribution is the product of uniform distributions on p c 1 and moreover p c 0 75 our proofs are based on a self similar graphical representation of the avalanches

boltzmann equation micro macro decompositions and positivity of shock profiles we introduce an elementary energy method for the boltzmann equation based on a decomposition of the equation into macroscopic and microscopic components the decomposition is useful for the study of time asymptotic stability of nonlinear waves the wave location is determined by the macroscopic equation the microscopic component has an equilibrating property the coupling of macroscopic and microscopic components gives rise naturally to the dissipations similar to those obtained by the chapman enskog expansion our main result is the establishment of the positivity of shock profiles for the boltzmann equation this is shown by the time asymptotic approach and the maximal principle for the collision operator

stable viscosity matrices for systems of conservation laws abstract a natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension u 1 f u x 0 u r m is studied these matrices are admissible in the sense that small amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system u t f u x v du x x as ifv 0 if d is in this class the class is determined by a linearized stability requirement the cauchy problem for the equation u 1 f u 0 u x vdu xx should be well posed in l 2 uniformly in v as v 0 previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion

l 1 stability of stationary discrete shocks the nonlinear stability in the l sup p norm p ge 1 of stationary weak discrete shocks for the lax friedrichs scheme approximating general m times m systems of nonlinear hyperbolic conservation laws is proved provided that the summations of the initial perturbations equal zero the result is proved by using a both weighted estimate and characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the lax friedrichs scheme 13 refs

bak sneppen type models and rank driven processes the bak sneppen model is a well known stochastic model of evolution that exhibits self organized criticality only a few analytical results have been established for it so far we report a surprising connection between bak sneppen type models and more tractable markov processes that evolve without any reference to an underlying topology specifically we show that in the case of a large number of species the long time behavior of the fitness profile in the bak sneppen model can be replicated by a model with a purely rank based update rule whose asymptotics can be studied rigorously

diffusive wave in the low mach limit for non viscous and heat conductive gas abstract the low mach number limit for one dimensional non isentropic compressible navier stokes system without viscosity is investigated where the density and temperature have different asymptotic states at far fields it is proved that the solution of the system converges to a nonlinear diffusion wave globally in time as mach number goes to zero it is remarked that the velocity of diffusion wave is proportional with the variation of temperature furthermore it is shown that the solution of compressible navier stokes system also has the same phenomenon when mach number is suitably small

galloping instability of viscous shock waves abstract motivated by physical and numerical observations of time oscillatory galloping spinning and cellular instabilities of detonation waves we study poincare hopf bifurcation of traveling wave solutions of viscous conservation laws the main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum preventing standard reduction to a finite dimensional center manifold we overcome this by direct lyapunov schmidt reduction using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap the key computation is a space time stability estimate on the transverse linearized solution operator reminiscent of duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves

efficient computation of analytic bases in evans function analysis of large systems in evans function computations of the spectra of asymptotically constant coefficient linearized operators of large systems a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices in the case that the invariant subspace is spectrally separated from its complementary invariant subspace we propose an efficient numerical implementation of a standard projection based algorithm of kato for which the key step is the solution of an associated sylvester problem this may be recognized as the analytic cousin of a c k algorithm developed by dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case for a one dimensional subspace it reduces essentially to an algorithm of bridges derks and gottwald based on path finding and continuation methods

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

l1nonlinear stability of traveling waves for a hyperbolic system with relaxation where x t r r the unknowns u v belong to r the function f f u is in c r and 0 is a fixed constant relaxation systems often arise in many physical situations let us recall for example gases not in thermodynamic equilibrium vk kinetic theory ce cip pi chromatography raa river flows traffic flows and more general waves wh more recently the study of a special class of hyperbolic systems with relaxation was developed in view of the numerical approximation of discontinuous solutions of conservation laws jx an the 2 2 relaxation hyperbolic systems of conservation laws were first analyzed by liu li2 who justified some nonlinear stability criteria for diffusion waves expansion waves and traveling waves article no 0180

mean field theory for a simple model of evolution a simple dynamical model for darwinian evolution on its slowest time scale is analyzed its mean field theory is formulated and solved a random neighbor version of the model is simulated as is a one dimensional version in one dimension the dynamics can be described in terms of a repetitious random walker and anomalous diffusion with exponent 0 4 in all cases the model self organizes to a robust critical attractor

stability of viscous detonations for majda s model abstract using analytical and numerical evans function techniques we examine the spectral stability of strong detonation wave solutions of a version of majda s scalar model for a reacting gas mixture with an arrhenius type ignition function we introduce an energy estimate to limit possible unstable eigenvalues to a compact region in the unstable complex half plane and we use a numerical approximation of the evans function to search for possible unstable eigenvalues in this region our results show for the parameter values tested that these waves are spectrally stable combining these numerical results with the pointwise green function analysis of lyng raoofi texier zumbrun g lyng m raoofi b texier k zumbrun pointwise green function bounds and stability of combustion waves j differential equations 233 2 2007 654 698 we conclude that these waves are nonlinearly stable this represents the first demonstration of nonlinear stability for detonation wave solutions of a majda type model without a smallness assumption notably our results indicate that for this simplified scalar model there does not occur either in a normal parameter range or in the limit of high activation energy hopf bifurcation to galloping or pulsating solutions as is observed in the full reactive navier stokes equations this answers in the negative for this model a question posed by majda as to whether such scalar detonation models capture this aspect of detonation behavior

linearized stability of extreme shock profiles for systems of conservation laws with viscosity abstract for a genuinely nonlinear hyperbolic system of conservation laws with added artificial viscosity we prove that traveling wave profiles for small amplitude extreme shocks the slowest and fastest are linearly stable to perturbations initial data chosen from certain spaces with weighted norm i e we show that the spectrum of the linearized equation lies strictly in the left half plane except for a simple eigenvalue at the origin due to phase translations of the profile the weight is used in components transverse to the profile where for an extreme shock the linearized equation is dominated by unidirectional convection

passage time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant in this paper we get some sufficient conditions for the finiteness or nonfiniteness of the passage time moments for nonnegative discrete parameter processes the developed criteria are closely connected with the well known results of foster for the ergodicity of markov chains and are given in terms of sub super martingales then as an application of the obtained results we get explicit conditions for the finiteness or nonfiniteness of passage time moments for reflected random walks in a quadrant with zero drift in the interior

stability of large amplitude viscous shock profiles of hyperbolic parabolic systems we establish nonlinear l1 h3 lp orbital stability 2 p with sharp rates of decay of large amplitude lax type shock profiles for a class of symmetric hyperbolic parabolic systems including compressible gas dynamics and magnetohydrodynamics mhd under the necessary conditions of strong spectral stability i e a stable point spectrum of the linearized operator about the wave transversality of the profile and hyperbolic stability of the associated ideal shock this yields in particular together with the spectral stability results of 50 the nonlinear stability of arbitrarily large amplitude shock profiles of isentropic navier stokes equations for a gamma law gas as 1 the first complete large amplitude stability result for a shock profile of a system with real i e partial viscosity a corresponding small amplitude result was established in 53 54 for general systems of kawashima class by a combination of kawashima type energy estimates and pointwise green function bounds where the small amplitude assumption was used only to close the energy estimates here under the mild additional assumption that hyperbolic characteristic speeds relative to the shock are not only nonzero but of a common sign we close the estimates instead by use of a goodman type weighted norm 25 26 designed to control estimates in the crucial hyperbolic modes