decay of metastable states with discrete dynamics we consider the escape from invariant sets of one dimensional piecewise linear maps which are additively disturbed by weak gaussian white noise the escape rates from point attractors and from strange invariant sets in the vicinity of the crisis at fully developed chaos are analytically determined and compared with results from numerical simulations both situations are combined resulting in a model with a point attractor which has a strange invariant set as basin boundary numerically a nonexponential decay of the attractor is found that can be described by a markovian three state model with transition rates known from the previous analysis

stochastic dynamics of the cubic map a study of noise induced transition phenomena the effects of finite amplitude additive noise on the dynamics generated by a one dimensional two parameter cubic map are considered the underlying deterministic system exhibits bistability and hysteresis and noise induced processes associated with these phenomena are studied if a bounded noise source is applied to this system trajectories may be confined to a finite region mechanisms are given for the merging transitions between different parts of this region and the eventual escape from it as the noise level is increased the noisy dynamics is also represented by an integral evolution operator with an equilibrium density function with finite support the operator s spectrum is determined as a function of map parameters and noise amplitude such noisy one dimensional maps can provide models for the study of noise induced phenomena described by stochastic differential equations

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

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

trace formulas for stochastic evolution operators weak noise perturbation theory periodic orbit theory is all effective tool for the analysis of classical and quantum chaotic systems in this paper we extend this approach to stochastic systems in particular to mappings with additive noise the theory is cast in the standard field theoretic formalism and weak noise perturbation theory written in terms of feynman diagrams the result is a stochastic analog of the next to leading h corrections to the gutzwiller trace formula with long time averages calculated from periodic orbits of the deterministic system the perturbative corrections are computed analytically and tested numerically on a simple 1 dimensional system

the influence of noise on the logistic model the behavior of the logistic system which is generated by the functionf x ax 1 x changes in an interesting way if it is perturbed by external noise it turns out that the chaotic behavior which was predicted by li and yorke for orbits of period 3 becomes visible and that a sequence of mergence transitions occurs at the critical parameter the change of the invariant probability density and the lyapunov exponents are examined numerically the power spectrum for the period 3 orbit for different fluctuations is calculated and a recursion formula for the time evolution of the probability density is presented as a discrete time analog of a chapman kolmogorov equation

diffusive wave in the low mach limit for compressible navier stokes equations abstract the low mach limit for 1d non isentropic compressible navier stokes flow whose density and temperature have different asymptotic states at infinity is rigorously justified the problems are considered on both well prepared and ill prepared data for the well prepared data the solutions of compressible navier stokes equations are shown to converge to a nonlinear diffusion wave solution globally in time as mach number goes to zero when the difference between the states at is suitably small in particular the velocity of diffusion wave is only driven by the variation of temperature it is further shown that the solution of compressible navier stokes system also has the same property when mach number is small which has never been observed before the convergence rates on both mach number and time are also obtained for the well prepared data for the ill prepared data the limit relies on the uniform estimates including weighted time derivatives and an extended convergence lemma and the difference between the states at can be arbitrary large in this case

stability of boundary layer and rarefaction wave to an outflow problem for compressible navier stokes equations under large perturbation abstract in this paper we investigate the large time behavior of solutions to an outflow problem for compressible navier stokes equations in 2003 kawashima nishibata and zhu s kawashima s nishibata p zhu asymptotic stability of the stationary solution to the compressible navier stokes equations in the half space comm math phys 240 2003 483 500 showed there exists a boundary layer i e stationary solution to the outflow problem and the boundary layer is nonlinearly stable under small initial perturbation in the present paper we show that not only the boundary layer above but also the superposition of a boundary layer and a rarefaction wave are stable under large initial perturbation the proofs are given by an elementary energy method

global classical solutions to the one dimensional compressible fluid models of korteweg type with large initial data abstract this paper is concerned with the global existence of classical solutions with large initial data away from vacuum to the cauchy problem of the one dimensional isothermal compressible fluid models of korteweg type with density dependent viscosity coefficient and capillarity coefficient the case when the viscosity coefficient and the capillarity coefficient for some parameters r is considered under some conditions on we first show the global existence of large solutions around constant states if the far fields of the initial data are the same while if the far fields of the initial data are different we prove the global stability of rarefaction waves with large strength here global stability means the initial perturbation can be arbitrarily large our analysis is based on the elementary energy method and the technique developed by y kanel 29

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

a simple fast adaptive algorithm for broad band null steering arrays a simple algorithm for adaptively steering the broadband null in a two element power inversion array which uses a tapped delay line filter as the processing element is presented and investigated by performing a simple signal transformation the algorithm constrains some of the filter weights so that a variable length delay line is approximated in the maximally flat sense by using the least mean square lms algorithm to adjust the weights controlling the length and gain of the delay line a broadband null is then adaptively formed to track and reject the broadband jammer as studied analytically and by using simulation results the convergence behavior of the algorithm is fast and depends only on the tolerable level of misadjustment noise its implementation complexity is comparable to that of using the lms algorithm directly

contact discontinuity with general perturbations for gas motions the contact discontinuity is one of the basic wave patterns in gas motions the stability of contact discontinuities with general perturbations for the navier stokes equations and the boltzmann equation is a long standing open problem general perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties in this paper we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the navier stokes equations and the boltzmann equation in a uniform way one of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of thus this reciprocal order of decay rates for the time evolution of the perturbation is essential to close the a priori estimate containing the uniform bounds of the l8 norm on the lower order estimate and then it gives the decay of the solution to the contact wave pattern

asymptotic stability of strong rarefaction waves for the compressible fluid models of korteweg type abstract this paper is concerned with the time asymptotic behavior toward strong rarefaction waves of solutions to one dimension compressible fluid models of korteweg type which governs the motions of the compressible fluids with internal capillarity assume that the corresponding riemann problem to the compressible euler system can be solved by rarefaction waves v r u r t x if the initial data is a small perturbation of an approximate rarefaction wave for v r u r t x we show that the corresponding cauchy problem admits a unique global smooth solution which tends to v r u r t x time asymptotically since we do not require the strength of the rarefaction waves to be small this result gives the nonlinear stability of strong rarefaction waves for the one dimensional compressible fluid models of korteweg type the analysis is based on the elementary l 2 energy method together with continuation argument

a fast null steering algorithm for linearly constrained adaptive arrays a fast null steering algorithm for adaptive arrays with a look direction constraint is investigated in this algorithm the least mean square lms algorithm is used to adaptively adjust the positions of the array response zeros one after another in a time multiplexed manner to track unknown jammers to ensure that the look direction response and the other zeros remain unchanged when a particular null is being updated the element signals are suitably transformed before the lms updating despite this signal transformation the new algorithm has almost the same implementation complexity as when the lms algorithm is employed directly as only one zero is being adaptively updated at any instant however with each null being independently steered the convergence behavior of the new algorithm is significantly faster and is almost independent of the external noise environment

nonlinear stability of rarefaction waves for the compressible navier stokes equations with large initial perturbation the expansion waves for the compressible navier stokes equations have recently been shown to be nonlinear stable the nonlinear stability results are called local stability or global stability depending on whether the h1 norm of the initial perturbation is small or not up to now local stability results have been well established however for global stability only partial results have been obtained the main purpose of this paper is to study the global stability of rarefaction waves for the compressible navier stokes equations for this purpose we introduce a positive parameter to in the construction of smooth approximations of the rarefaction wave solutions for the compressible euler equations so that the quantity ty 6 denotes the strength of the rarefaction waves is sufficiently large to control the growth induced by the nonlinearity of the system and the interaction of waves from different families then by using the energy method together with the continuation argument we obtain some nonlinear stability results provided that the initial perturbation satisfies certain growth conditions as oo notice that the assumption that the quantity can be chosen to be sufficiently large implies that either the strength of the rarefaction waves is small or the rarefaction waves of different families are separated far enough initially

zero dissipation limit to rarefaction waves for the one dimensional navier stokes equations of compressible isentropic gases we study the zero dissipation limit problem for the one dimensional navier stokes equations of compressible isentropic gases in the case that the corresponding euler equations have rarefaction wave solutions we prove that the solutions of the navier stokes equations with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity vanishes uniformly away from the initial discontinuities in the case that either the effects of initial layers are ignored or the rarefaction waves are smooth we then obtain a rate of convergence which is valid uniformly for all time our method of proof consists of a scaling argument and elementary energy analysis based on the underlying wave structure 1993 john wiley sons inc

adaptive zero tracking complemented by lms weight update for guaranteed convergence a low complexity zero tracking algorithm with fast and guaranteed convergence behavior is proposed and investigated the algorithm implements a zero tracking algorithm and a least mean square lms weight update algorithm in parallel with the former adjusting the zeros of the array in a time multiplexed manner to achieve fast asymptotic and tracking behavior while the latter improves the initial transient behavior of the algorithm and guarantees its convergence to the global optimum by comparing their output powers the relative performances of the two component algorithms are monitored and re initialization of one algorithm by the other may occur periodically this gives the algorithm both fast and guaranteed convergence behavior even though the zeros are directly available and the implementation complexity is only two times that for the conventional lms algorithm

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

quasipotentials for simple noisy maps with complicated dynamics the theory of nonequilibrium potentials or quasipotentials is a physically motivated approach to small random perturbations of dynamical systems leading to exponential estimates of invariant probabilities and mean first exit times in the present article we develop the mathematical foundation of this theory for discrete time systems following and extending the work of freidlin and wentzell and kifer we discuss strategies for calculating and estimating quasipotentials and show their application to one dimensionals unimodal maps the method proves to be especially suited for describing the noise scaling behavior of invariant probabilities e g for the map occurring as the limit of the feigenbaum period doubling sequence we show that the method allows statements about the scaling behavior in the case of localized noise too which does not originally lie within the scope of the quasipotential formalism

adaptive beam forming using a cascade configuration a new technique for adaptive beam forming based on the davies cascade array configuration 11 is presented first of all it is shown that an n element linear array can be reconfigured as a number of subarrays of two elements each this two element array is capable of steering one null in an arbitrary direction while maintaining a desired gain in some other prespecified look direction to adaptively perform this function at any cascade stage the inverse of a 2 2 correlation matrix of the signal vector at a subarray is computed and substituted in the expression for the constrained optimum weight vector similar processing is performed at subsequent stages of the cascade configuration at each stage of adaptation a noise source contributing maximum power at that stage is nulled out simulation studies carried out verify the validity of this approach and demonstrate the fast convergence property of the proposed technique

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

convergence to the superposition of rarefaction waves and contact discontinuity for the 1 d compressible navier stokes korteweg system abstract this paper is concerned with the vanishing capillarity viscosity limit for the one dimensional compressible navier stokes korteweg system to the riemann solution of the euler system that consists of the supposition of two rarefaction waves and a contact discontinuity it is shown that there exists a family of smooth solutions to the compressible navier stokes korteweg system which converge to the riemann solution away from the initial time t 0 and the contact discontinuity located at x 0 as the coefficients of capillarity viscosity and heat conductivity tend to zero moreover a uniform convergence rate in terms of the above physical parameters is also obtained here the strengths of both the rarefaction waves and the contact discontinuity are not required to be small

decay rate to contact discontinuities for the 1 d compressible navier stokes system abstract we revisit the classical work of huang matsumura xin 9 for contact wave of the one dimensional compressible navier stokes system under the zero mass condition the large time asymptotic stability of a contact wave pattern with a uniform convergence rate 1 t 1 4 was proved by huang matsumura xin in this paper huang matsumura xin s convergence rate is improved to 1 t 5 8 ln 1 2 2 t by using a new poincare type inequality and a detailed energy analysis our work is motivated by a problem arising in 9

stability of contact discontinuities for the 1 d compressible navier stokes equations in this paper we study the large time asymptotic behavior of solutions of the one dimensional compressible navier stokes system toward a contact discontinuity which is one of the basic wave patterns for the compressible euler equations it is proved that such a weak contact discontinuity is a metastable wave pattern in the sense introduced in 24 for the 1 d compressible navier stokes system for polytropic fluid by showing that a viscous 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 is nonlinearly stable with a uniform convergence rate provided that the initial excess mass is zero this result is proved by an elaborate combination of elementary energy estimates with a weighted characteristic energy estimate which makes full use of the underlying structure of the viscous contact wave

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

noisy one dimensional maps near a crisis i weak gaussian white and colored noise we study one dimensional single humped maps near the boundary crisis at fully developed chaos in the presence of additive weak gaussian white noise by means of a new perturbation like method the quasi invariant density is calculated from the invariant density at the crisis in the absence of noise in the precritical regime where the deterministic map may show periodic windows a necessary and sufficient condition for the validity of this method is derived from the quasi invariant density we determine the escape rate which has the form of a scaling law and compares excellently with results from numerical simulations we find that deterministic transient chaos is stabilized by weak noise whenever the maximum of the map is of orderz 1 finally we extend our method to more general maps near a boundary crisis and to multiplicative as well as colored weak gaussian noise within this extended class of noises and for single humped maps with any fixed orderz 0 of the maximum in the scaling law for the escape rate both the critical exponents and the scaling function are universal

noisy one dimensional maps near a crisis ii general uncorrelated weak noise the escape rate for one dimensional noisy maps near a crisis is investigated a previously introduced perturbation theory is extended to very general kinds of weak uncorrelated noise including multiplicative white noise as a special case for single humped maps near the boundary crisis at fully developed chaos an asymptotically exact scaling law for the rate is derived it predicts that transient chaos is stabilized by basically any noise of appropriate strength provided the maximum of the map is of sufficiently large order a simple heuristic explanation of this effect is given the escape rate is discussed in detail for noise distributions of levy dichotomous and exponential type in the latter case the rate is dominated by an exponentially leading arrhenius factor in the deep precritical regime however the preexponential factor may still depend more strongly than any power law on the noise strength

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

rates and mean first passage times the relation between mean first passage times t and transition rates f in noisy dynamical systems with metastahle states is investigated it is shown that the inverse mean first passage time to the separatrix of the noiseless system may deviate from twice the rate not only because in general the deterministic separatrix is not the locus in the state space from which a noisy trajectory goes to either side with equal probability a further cause of a deviation from the often assumed relation ft 1 2 between rates and mean first passage times is given if the noisy dynamics is discontinuous i e shows jumps with finite probability then the value of the splitting probability at the separatrix does not fix the value of ft since the system need not visit the separatrix during a transition from one to the other side most important for discontinuous processes the deviation from the ft rule survives even in the weak noise limit a mathematical relation for the product of the rate and the mean first passage time is proposed for markovian processes and numerically confirmed for a particular one dimensional noisy map

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

properties of fully developed chaos in one dimensional maps we consider single humped symmetric one dimensional maps generating fully developed chaotic iterations specified by the property that on the attractor the mapping is everywhere two to one to calculate the probability distribution function and in turn the lyapunov exponent and the correlation function a perturbation expansion is developed for the invariant measure besides deriving some general results we treat several examples in detail and compare our theoretical results with recent numerical ones furthermore a necessary condition is deduced for the probability distribution function to be symmetric and an effective functional iteration method for the measure is introduced for numerical purposes

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

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

suppression of deterministic diffusion by noise a one dimensional discrete time dynamics near the onset of deterministic diffusion additively coupled to weak gaussian white noise is studied the diffusion coefficient is related to the escape rate out of an interval and a method for the calculation of this rate is introduced which unlike previous investigations fully takes into account all relevant escape paths it is found that the diffusion coefficient of the deterministic dynamics may be considerably reduced by the noise in agreement with numerical simulations it is shown that certain fundamental problems in the concept of deterministic diffusion without noise do not affect the validity of our result

global stability of the rarefaction wave of a one dimensional model system for compressible viscous gas this paper is concerned with the asymptotic behavior toward the rarefaction wave of the solution of a one dimensional barotropic model system for compressible viscous gas we assume that the initial data tend to constant states atx respectively and the riemann problem for the corresponding hyperbolic system admits a weak continuous rarefaction wave if the adiabatic constant satisfies 1 2 then the solution is proved to tend to the rarefaction wave ast under no smallness conditions of both the difference of asymptotic values atx and the initial data the proof is given by an elementaryl 2 energy method

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

stochastic perturbations in open chaotic systems random versus noisy maps we investigate the effects of random perturbations on fully chaotic open systems perturbations can be applied to each trajectory independently white noise or simultaneously to all trajectories random map we compare these two scenarios by generalizing the theory of open chaotic systems and introducing a time dependent conditionally map invariant measure for the same perturbation strength we show that the escape rate of the random map is always larger than that of the noisy map in random maps we show that the escape rate ensuremath kappa and dimensions d of the relevant fractal sets often depend nonmonotonically on the intensity of the random perturbation we discuss the accuracy bias and precision variance of finite size estimators of ensuremath kappa and d and show that the improvement of the precision of the estimations with the number of trajectories n is extremely slow ensuremath propto 1 mathrm ln n we also argue that the finite size d estimators are typically biased general theoretical results are combined with analytical calculations and numerical simulations in area preserving baker maps

asymptotic stability of combination of viscous contact wave with rarefaction waves for one dimensional compressible navier stokes system we are concerned with the large time behavior of solutions of the cauchy problem to the one dimensional compressible navier stokes system for ideal polytropic fluids where the far field states are prescribed when the corresponding riemann problem for the compressible euler system admits the solution consisting of contact discontinuity and rarefaction waves it is proved that for the one dimensional compressible navier stokes system the combination wave of a viscous contact wave which corresponds to the contact discontinuity with rarefaction waves is asymptotically stable provided the strength of the combination wave is suitably small this result is proved by using elementary energy methods