the green s function and large time behavior of solutions for the one dimensional boltzmann equation we study the pointwise behavior of the green s function of the boltzmann equation our results reveal the particle and fluid aspects of the equation the particle aspect is represented by singular waves these waves are carried by transport equations and dominate the short time behavior of the solution we devise a picard type iteration for constructing the increasingly regular particlelike waves the fluidlike waves reveal the dissipative behavior of the type of navier stokes equations as usually seen by the chapman enskog expansion these waves are constructed as part of the long wave expansion in the spectrum of the fourier mode for the space variable the fluidlike waves represent the long time behavior of the solution as an application we obtain the pointwise description of the large time behavior of the convergence to the global maxwellian when the initial perturbation is not necessarily smooth in our analysis of the exchanges of the microscopic velocity decay and space decay we make essential use of the hard sphere models c 2004 wiley periodicals inc contents

splitting scheme for the stability of strong shock profile abstract in this paper we use the laplace transform and dirichlet neumann map to give a systematical scheme to study the small wave perturbation of general shock profile with general amplitude here we use certain non classical shock waves for a rotationally invariant system of viscous conservation laws to demonstrate this scheme we derive an explicit solution and show that it converges pointwise to another over compressive profile exponentially when the perturbations of the initial data to a given over compressive shock profile are sufficiently small

existence and stability of stationary profiles of the lw scheme in this paper we study the behavior of difference schemes approximating solutions with shocks of scalar conservation laws when a difference scheme introduces artificial numerical diffusion for example the lax friedrichs scheme we experience smearing of the shocks whereas when a scheme introduces numerical dispersion for example the lax wendroff scheme we experience oscillations which decay exponentially fast on both sides of the shock in his dissertation gray jennings studied approximation by monotone schemes these contain artificial viscosity and are first order accurate they are known to be contractive in the sense of any lp norm jennings showed existence and l1 stability of traveling discrete smeared shocks for such schemes here we study similar questions for the lax wendroff scheme without artificial viscosity this is a nonmonotone second order accurate scheme we prove existence of a one parameter family of stationary profiles we also prove stability of these profiles for small perturbations in the sense of a suitably weighted l2 norm the proof relies on studying the linearized lax wendroff scheme

nonlinear stability of large amplitude viscous shock wave for general viscous gas abstract in the present paper it is shown that the large amplitude viscous shock wave is nonlinearly stable for isentropic navier stokes equations in which the pressure could be general and includes law and the viscosity coefficient is a smooth function of density the strength of shock wave could be arbitrarily large the proof is given by introducing a new variable which can formulate the original system into a new one and the elementary energy method introduced in 21

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

elimination of first order errors in shock calculations first order errors downstream of shocks have been detected in computations with higher order shock capturing schemes in one and two dimensions based on a matched asymptotic expansion analysis we show how to modify the artificial viscosity and raise the order of accuracy

global existence of plasma ion sheaths and their dynamics when a negative high voltage pulse is applied to a target material immersed in a plasma a boundary layer sheath forms around the target and the motion of cold ions in the plasma is governed by the euler poisson system in this paper by simplifying the euler poisson system on suitable physical regimes we present a theory for the existence and dynamics of time dependent sheaths with planar cylindrical and spherical symmetry for the construction of ion sheaths we employ the method of characteristics and study the dynamic behavior of a plasma sheath edge based on odes which are formally derived from the euler poisson system

existence and stability of planar diffusion waves for 2 d euler equations with damping abstract to study the non linear stability of a non trivial profile for a multi dimensional systems of gas dynamics the combination of the green function on estimating the lower order derivatives and the energy method for the higher order derivatives is shown to be not only useful but sometimes maybe also essential in this paper we study the stability of a planar diffusion wave for the isentropic euler equations with damping in two dimensional space by introducing an approximate green function for the linearized equations around the planar diffusion wave and by applying the energy method we prove the global existence and the l 2 convergence rate of the solution when the initial data is a small perturbation of the planar diffusion wave the decay rates of the perturbation and its lower order spatial derivatives obtained are optimal in the l 2 norm furthermore the constructed approximate green function in this paper can be used for the pointwise and the l p estimates of the solutions concerned in fact the approach by combining of the green function and energy method can be applied to other system especially when the derivatives of the coefficients in the system have certain time decay properties

the convergence rate of finite difference schemes in the presence of shocks finite difference approximations generically have cal o 1 pointwise errors close to a shock we show that this local error may effect the smooth part of the solution such that only first order is achieved even for formally higher order methods analytic and numerical examples of this form of accuracy are given we also show that a converging method will have the formal order of accuracy in domains where no characteristics have passed through a shock

l 2 stability estimates for shock solutions of scalar conservation laws using the relative entropy method we consider scalar nonviscous conservation laws with strictly convex flux in one spatial dimension and we investigate the behavior of bounded l2 perturbations of shock wave solutions to the riemann problem using the relative entropy method we show that up to a time dependent translation of the shock the l2 norm of a perturbed solution relative to the shock wave is bounded above by the l2 norm of the initial perturbation

analysis of first order errors in shock calculations in two space dimensions numerical computations show that solutions of hyperbolic conservation laws obtained by second or higher order shock capturing methods in many cases are only first order accurate downstream of shocks see e g m h carpenter and j h casper aiaa j 37 1999 pp 1072 1079 we use matched asymptotic expansions to analyze the degeneration in order of accuracy for stationary solutions of hyperbolic conservation laws in two space dimensions

broadwell model and conservative supersonic boundary in this paper we consider an initial boundary value problem for the broadwell model with a supersonic physical boundary here we pose a conservative boundary condition which preserves the total number of particles we show that the solution converges pointwise to a boundary layer exponentially when the perturbations of the initial data to the equilibrium state are sufficiently small by using green s function as established in lan et al netw heterog media 1 167 183 2006 weighted energy estimates and by constructing a pair of anti derivatives to convert the conservative boundary condition into a dissipative boundary condition with conservation laws together with an a priori chosen boundary layer

stability of discrete shocks for difference approximations to systems of conservation laws the asymptotic stability of weak discrete stationary shocks for systems of conservation laws in one space dimension is proved the difference approximation should be conservative dissipative and kth order accurate in space with odd k the problem is considered in a finite interval vert x vert le ell with appropriate boundary conditions where ell is large compared with the width of the shock layer varepsilon 1 vert u r u l vert 1 k the proof is based on the assumption that the corresponding continuous shocks for the scalar problem u t uu x i partial x k 1 u are stable the latter is known to be true for k 1 and k 3

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 behavior near transition fronts for equations of generalized cahn hilliard form we consider the asymptotic behavior of perturbations of standing wave solutions arising in evolutionary pde of generalized cahn hilliard form in one space dimension such equations are well known to arise in the study of spinodal decomposition a phenomenon in which the rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into its two components with their concentrations separated by sharp transition layers motivated by work of bricmont kupiainen and taskinen 5 we regard the study of standing waves as an interesting step toward understanding the dynamics of these transitions a critical feature of the cahn hilliard equation is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from standing wave solutions establishing phase asymptotic orbital stability for initial perturbations decaying with appropriate algebraic rate

computational considerations for the simulation of shock induced sound the numerical study of aeroacoustic problems places stringent demands on the choice of a computational algorithm because it requires the ability to propagate disturbances of small amplitude and short wavelength the demands are particularly high when shock waves are involved because the chosen algorithm must also resolve discontinuities in the solution the extent to which a high order accurate shock capturing method can be relied upon for aeroacoustics applications that involve the interaction of shocks with other waves has not been previously quantified such a study is initiated in this work a fourth order accurate essentially nonoscillatory eno method is used to investigate the solutions of inviscid compressible flows with shocks the design order of accuracy is achieved in the smooth regions of a steady state quasi one dimensional test case however in an unsteady test case only first order results are obtained downstream of a sound shock interaction the difficulty in obtaining a globally high order accurate solution in such a case with a shock capturing method is demonstrated through the study of a simplified linear model problem some of the difficult issues and ramifications for aeroacoustic simulations of flows with shocks that are raised by these results are discussed

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

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

a remark on numerical errors downstream of slightly viscous shocks lower order errors downstream of a shock layer have been detected in computations with nonconstant solutions when using higher order shock capturing schemes in one and two dimensions b engquist and b sjogreen siam j numer anal 35 1998 pp 2464 2485 by analyzing the steady state solution of slightly viscous hyperbolic systems of conservation laws we find that the solution can have an cal o h dependence downstream of a shock layer although the viscous term in that region is of cal o h 2 numerical examples illustrate the analysis

computations of slowly moving shocks computations of slowly moving shocks by shock capturing schemes may generate oscillations that appear as a wavy tail attached to the shock front these oscillations are generated already by first order schemes but become more pronounced in higher order schemes due to their lower dissipation we focus on two first order schemes which seem to exhibit different behaviors i the first order upwind uw scheme which generates strong oscillations and ii the lax friedrichs scheme which appears not to generate any disturbances at all a key observation is that in the uw case the numerical viscosity in the shock family vanishes inside the slow shock layer simple scaling arguments show that third order effects on the solution may no longer be neglected we derive the third order modified equation for the uw scheme and regard the oscillatory solution as a traveling wave solution of the parabolic modified equation plus a small perturbation we then look at the governing equation for the perturbation which points to a plausible mechanism by which postshock oscillations are generated it contains a third order source term that becomes significant inside the shock layer and a nonlinear coupling term which projects the perturbation on all characteristic fields including those not associated with the shock family

l p convergence rates of planar waves for multi dimensional euler equations with damping abstract in this paper the l p convergence rates of planar diffusion waves for multi dimensional euler equations with damping are considered the analysis relies on a newly introduced frequency decomposition and green function based energy method it is a combination of the l p estimate on the low frequency component by using an approximate green function and l 2 estimate on the high frequency component through the energy method by noticing that the low frequency component in the approximate green function has the algebraic decay which governs the large time behavior while the high frequency component has the exponential decay but with singularity their combination leads to a global algebraic decay estimate to use the decay property only of the low frequency component in the approximate green function avoids the singularity in the high frequency component so that it simplifies and improves the previous works on this system this new approach of the combination of the green function and energy method through the frequency decomposition can also be applied to the hyperbolic parabolic systems satisfying the kawashima condition and also the systems whose derivatives of the coefficients have suitable time decay properties

asymptotic behaviors of solutions of quasilinear hyperbolic equations with linear damping ii this paper as a follow up to the recent work of k nishihara 1997 j differential equations133 384 395 is concerned with the asymptotic behaviors of the solution of quasilinear hyperbolic equation with linear damping vtt a vx x vt 0 a v 0forallv r which satisfies the following prescribed initial condition v t x vt t x t 0 v0 x v1 x x r compared with the results obtained by k nishihara the main novelties of our present paper lie in the following first we recover all the results obtained by k nishihara under some other smallness assumptions on the initial data v0 x v1 x which are much weaker than those needed in k nishihara s arguments second under some additional assumptions on the nonlinear function a v and on the initial data v0 x v1 x when 0 0 1 0 we succeed in proving that t x is still an asymptotic profile of v t x and this answers partially the open problem left by k nishihara

algebras with one operation including poisson and other lie admissible algebras abstract we investigate algebras with one operation we study when these algebras form a monoidal category and analyze koszulness and cyclicity of the corresponding operads we also introduce a new kind of symmetry for operads the dihedrality responsible for the existence of dihedral cohomology the main trick which we call the polarization will be used to represent an algebra with one operation without any specific symmetry as an algebra with one commutative and one anticommutative operations we will try to convince the reader that this change of perspective might sometimes lead to new insights and results this point of view was used by livernet and loday to introduce a one parameter family of operads whose specialization at 0 is the operad for poisson algebras while at a generic point it equals the operad for associative algebras we study this family and explain how it can be used to interpret the deformation quantization product in a neat and elegant way

exact decimation transformation with mean field approximation a new method is developed for calculating the critical temperature and exponents of spin systems based on the exact decimation transformation and a mean field approximation calculations of both exponents and critical couplings are much improved even in the lowest approximation when compared with other renormalisation group methods

lie admissible algebras and operads abstract a lie admissible algebra gives a lie algebra by anticommutativity in this work we describe remarkable types of lie admissible algebras such as vinberg algebras pre lie algebras or lie algebras we compute the corresponding binary quadratic operads and study their duality considering lie algebras as lie admissible algebras we can define for each lie algebra a cohomology with values in an lie admissible module this leads to the study some deformations of lie algebras in the classes of lie admissible algebras

thermodynamic functions and critical properties from a cluster decimation approximation a simple hybrid of real space renormalisation group theory and finite lattice calculations is described the method is based on a lattice restructuring transformation with the criterion that the free energies of the original and restructured lattices be equal the authors approximate the free energy per spin of the infinite lattice by an exact evaluation of the partition function for a small cluster as a result of this transformation the coupling constant of the restructured lattice is found to be temperature dependent the method is illustrated by studying the ising model in two and three dimensions and two dimensional potts models the calculations involve little more than is required for a migdal kadanoff transformation yet the critical properties are found to be significantly improved in comparison detailed comparisons are also made with the global thermodynamic properties calculated within this approximation free energy internal energy specific heat spontaneous magnetisation and exact or best known results for the models the simplicity of the method allows for the study of more complex lattice models and several possible applications are discussed

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

pointwise green function bounds for shock profiles of systems with real viscosity following the pointwise semigroup approach of zh mz 1 we establish sharp pointwise green function bounds and consequent linearized stability for viscous shock profiles of general hyperbolic parabolic systems of conservation laws of dissipative type under the necessary assumptions z 1 z 3 z 4 of spectral stability i e stable point spectrum of the linearized operator about the wave transversality of the connecting profile and hyperbolic stability of the corresponding ideal shock of the associated inviscid system with no additional assumptions on the structure or strength of the shock these bounds are used in a companion paper mz 2 to establish nonlinear stability of small amplitude lax shocks of symmetrizable hyperbolic parabolic systems

nonlinear stability of strong detonation waves for a dissipative model we study the asymptotic behavior of solutions to a combustion model which contains the effect of convection viscosity chemical reaction and diffusion it is proved that if a perturbation of a travelling strong detonation wave is small then the solution to this system converges to a strong detonation wave which is the original one with a shift as t infinity the rate of convergence is also estimated c 1999 academic press

dynamics and stability of a weak detonation wave one dimensional weak detonation waves of a basic reactive shock wave model are proved to be nonlinearly stable i e initially perturbed waves tend asymptotically to translated weak detonation waves this model system was derived as the low mach number limit of the one component reactive navier stokes equations by majda and roytburd siam j sci stat comput 43 1086 1118 1983 and its weak detonation waves have been numerically observed as stable the analysis shows in particular the key role of the new nonlinear dynamics of the position of the shock wave the shock translation solves a nonlinear integral equation obtained by green s function techniques and its solution is estimated by observing that the kernel can be split into a dominating convolution operator and a remainder the inverse operator of the convolution and detailed properties of the traveling wave reduce by monotonicity the remainder to a small l1 perturbation

pointwise green s function bounds for multidimensional scalar viscous shock fronts we determine detailed pointwise bounds for the green s function of the linearized operator about a multidimensional scalar viscous shock front these extend the pointwise semigroup methods introduced by howard and zumbrun in the one dimensional case to multidimensions sharpening lp estimates obtained by goodman and miller using a weighted norm approach moreover our results apply to shocks of arbitrary strength as previous results did not as described in a companion paper the bounds we obtain are sufficient to give a straightforward treatment of the nonlinear lp asymptotic behavior of the front under small perturbation the analysis of the multidimensional case involves several new features not found in the one dimensional case concerned with the geometry of propagating signals

the broadwell model with a subsonic boundary abstract in this paper we focus on the time asymptotic behavior of an initial boundary value problem ibvp for the broadwell model with a subsonic physical boundary by using the green s function for the initial problem established in c y lan h e lin s h yu the green s functions for the broadwell model in half space problem netw heterog media 1 1 2006 and the weighted energy estimates we construct the green s function for ibvp and show that the solution converges pointwise to the equilibrium state when the perturbations are sufficiently small

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

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

pointwise green s function estimates toward stability for multidimensional fourth order viscous shock fronts abstract for the case of multidimensional viscous conservation laws with fourth order smoothing only we develop detailed pointwise estimates on the green s function for the linear fourth order convection equation that arises upon linearization of the conservation law about a viscous planar wave solution as in previous analyses in the case of second order smoothing our estimates are sufficient to establish that spectral stability implies nonlinear stability though the full development of this result will be considered in a companion paper

existence of relaxation shock profiles for hyperbolic conservation laws in this paper we prove that like viscosity relaxation can also smooth weak shocks for strict hyperbolic conservation laws as equilibium systems of hyperbolic relaxation systems under some natural structural conditions these conditions were derived previously by yong from stability considerations and hence are entirely analogous to the majda pego criterion of the viscous case the proof involves a new parametrization of hugoniot curves and a delicate analysis of the algebraic structure of relaxation systems as well as construction of an appropriate center manifold

existence of discrete shock profiles of a class of monotonicity preserving schemes for conservation laws when shock speed s times t x is rational the existence of solutions of shock profile equations on bounded intervals for monotonicity preserving schemes with continuous numerical flux is proved a sufficient condition under which the above solutions can be extended to j implying the existence of discrete shock profiles of numerical schemes is provided a class of monotonicity preserving schemes including all monotonicity preserving schemes with c 1 numerical flux functions the second order upwinding flux based muscl scheme the second order flux based muscl scheme with lax friedrichs splitting and the godunov scheme for scalar conservation laws are found to satisfy this condition thus the existence of discrete shock profiles for these schemes is established when s t x is rational

an investigation of the internal structure of shock profiles for shock capturing schemes the theoretical understanding of discrete shock transitions obtained by shock capturing schemes is very incomplete previous experimental studies indicate that discrete shock transitions obtained by shock capturing schemes can be modeled by continuous functions so called continuum shock profiles however the previous papers have focused on linear methods we have experimentally studied the trajectories of discrete shock profiles in phase space for a range of different high resolution shock capturing schemes including riemann solver based flux limiter methods high resolution central schemes and eno type methods in some cases no continuum profiles exists however in these cases the point values in the shock transitions remain bounded and appear to converge toward a stable limit cycle the possibility of such behavior was anticipated in bultelle grassin and serre 1998 but no specific examples or other evidence of this behavior have previously been given in other cases our results indicate that continuum shock profiles exist but are very complicated we also study phase space orbits with regard to post shock oscillations

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

lp convergence rate to nonlinear diffusion waves for p system with damping abstract in this paper we study the p system with frictional damping and show that the solutions time asymptotically tend to the nonlinear diffusion waves governed by the classical darcy s law by introducing an approximate green function we obtain the optimal l p 2 p convergence rate of the solution which is a perturbation of the nonlinear diffusion wave to the hyperbolic system

stability of small amplitude shock profiles of symmetric hyperbolic parabolic systems combining pointwise green s function bounds obtained in a companion paper 36 with earlier spectral stability results obtained in 16 we establish nonlinear orbital stability of small amplitude lax type viscous shock profiles for the class of dissipative symmetric hyperbolic parabolic systems identified by kawashima 20 notably including compressible navier stokes equations and the equations of magnetohydrodynamics obtaining sharp rates of decay in lp with respect to small l1 h3 perturbations 2 p our analysis extends and somewhat refines the approach introduced in 35 to treat stability of relaxation profiles 2004 wiley periodicals inc

pointwise green s function approach to stability for scalar conservation laws we study the pointwise behavior of perturbations from a viscous shock solution to a scalar conservation law obtaining an estimate independent of shock strength we find that for a perturbation with initial data decaying algebraically or slower the perturbation decays in time at the rate of decay of the integrated initial data in any l p norm p 1 stability in any l p norm is a direct consequence the approach taken is that of obtaining pointwise estimates on the perturbation through a duhamel s principle argument that employs recently developed pointwise estimates on the green s function for the linearized equation c 1999 john wiley sons inc

stability of undercompressive shock profiles abstract using a simplified pointwise iteration scheme we establish nonlinear phase asymptotic orbital stability of large amplitude lax undercompressive overcompressive and mixed under overcompressive type shock profiles of strictly parabolic systems of conservation laws with respect to initial perturbations u 0 x e 0 1 x 3 2 in c 0 e 0 sufficiently small under the necessary conditions of spectral and hyperbolic stability together with transversality of the connecting profile this completes the program initiated by zumbrun and howard in k zumbrun p howard pointwise semigroup methods and stability of viscous shock waves indiana univ math j 47 4 1998 741 871 extending to the general undercompressive case results obtained for lax and overcompressive shock profiles in a szepessy z xin nonlinear stability of viscous shock waves arch ration mech anal 122 1993 53 103 t p liu pointwise convergence to shock waves for viscous conservation laws comm pure appl math 50 11 1997 1113 1182 k zumbrun p howard pointwise semigroup methods and stability of viscous shock waves indiana univ math j 47 4 1998 741 871 k zumbrun refined wave tracking and nonlinear stability of viscous lax shocks methods appl anal 7 2000 747 768 m r raoofi l 1 asymptotic behavior of perturbed viscous shock profiles thesis indiana univ 2004 c mascia k zumbrun pointwise green s function bounds and stability of relaxation shocks indiana univ math j 51 4 2002 773 904 c mascia k zumbrun stability of small amplitude shock profiles of symmetric hyperbolic parabolic systems comm pure appl math 57 7 2004 841 876 c mascia k zumbrun pointwise green s function bounds for shock profiles with degenerate viscosity arch ration mech anal 169 3 2003 177 263 c mascia k zumbrun stability of large amplitude shock profiles of hyperbolic parabolic systems arch ration mech anal 172 1 2004 93 131 c mascia k zumbrun stability of large amplitude shock profiles of general relaxation systems siam j math anal in press and for special weakly coupled respectively scalar diffusive dispersive undercompressive profiles in t p liu k zumbrun nonlinear stability of an undercompressive shock for complex burgers equation comm math phys 168 1 1995 163 186 t p liu k zumbrun on nonlinear stability of general undercompressive viscous shock waves comm math phys 174 2 1995 319 345 respectively p howard k zumbrun pointwise estimates for dispersive diffusive shock waves arch ration mech anal 155 2000 85 169 in particular together with spectral results of k zumbrun dynamical stability of phase transitions in the p system with viscosity capillarity siam j appl math 60 2000 1913 1924 our results yield nonlinear stability of large amplitude undercompressive phase transitional profiles near equilibrium of slemrod s model m slemrod admissibility criteria for propagating phase boundaries in a van der waals fluid arch ration mech anal 81 4 1983 301 315 for van der waal gas dynamics or elasticity with viscosity capillarity

convergence to strong nonlinear diffusion waves for solutions of p system with damping this paper is concerned with the p system with frictional damping and our main purpose is two fold first we show that for a certain class of given large initial data nu 0 x u 0 x the cauchy problem 1 1 1 2 admits a unique global smooth solution nu t x u t x and such a solution tends time asymptotically at the optimal l p 2 less than or equal to p less than or equal to infinity decay rates to the corresponding nonlinear diffusion wave over bar t x u over bar t x governed by the classical darcy s law provided that the corresponding prescribed initial error function v 0 x u 0 x ties in h 3 x h 2 r boolean and l 1 x l 1 r compared with former results in this direction obtained by l hsiao and t p liu 1992 comm math phys 143 599 605 k nishihara 1996 j differential equations 131 171 188 and k nishihara w k wang and t yang 2000 j differential equations 161 191 218 our main novelty lies in the facts that the nonlinear diffusion wave over bar t x u over bar t x need not to be weak and v 0 x u 0 x can be chosen arbitrarily large secondly we show that the nonlinear diffusion waves over bar t x u over bar t x are nonlinear stable provided that the strength or the nonlinear diffusion waves is weak and that the initial disturbance v 0 x u 0 x satisfies the assumption that parallel tov 0xx x parallel to l infinity parallel tou 0x x parallel to l infinity is sufficiently small we also show that the smallness assumption imposed oil the strength of the diffusion waves is a necessary condition to guarantee the nonlinear stability result and compared with the corresponding results obtained by l hsiao and t p liu 1992 comm math phys 143 599 605 the smallness conditions we imposed on the initial disturbance are much more weaker c 2001 academic press

the pointwise estimates of solutions for euler equations with damping in multi dimensions we study the time asymptotic behavior of solutions for the isentropic euler equations with damping in multi dimensions the global existence and pointwise estimates of the solutions are obtained furthermore we obtain the optimal lp 1 p convergence rate of the solution when it is a perturbation of a constant state our approach is based on a detailed analysis of the green function of the linearized system and some energy estimates

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

multidimensional viscous shocks i degenerate symmetrizers and long time stability received by the editors september 18 2002 2000 mathematics subject classification primary 35l60 secondary 35b35 35b45 35k55 35l65 76l05

pointwise convergence to knudsen layers of the boltzmann equation in this paper green s functions for the boltzmann equation around a global maxwellian are used to construct the non characteristic nonlinear knudsen layers as well as their time asymptotic stability furthermore the detailed pointwise structures nonlinear wave couplings and wave interactions with boundary are studied

asymptotic behavior of solutions to a system of viscous conservation laws related to a phase transition problem we study the asymptotic behavior of solutions for a system of viscous conservation laws with discontinuous initial data we discuss mainly the case where the system without the viscosity term is of hyperbolic elliptic mixed type this problem is related to a phase transition problem we study the initial value problem and show the decay rates of solutions to piecewise constant states where two phases coexist the modification necessary for the hyperbolic case is also discussed

a simple improvement of the migdal kadanoff renormalisation group scheme we construct a family of renormalisation group transformations r both for the ising model on a triangular lattice and for the ising model with checkerboard interactions on a square lattice for all parameter values a r implies a lower bound to the free energy the usual migdal kadanoff scheme is recovered for a 0 however the optimal value for the thermal scaling index yt is obtained for a a f 0 with dyt a aul 0

initial boundary value problem for one dimensional wave solutions of the boltzmann equation tai ping liu and his collaborator shih hsien yu solve the initial boundary value problem for the boltzmann equation because of the inclusion of microscopic velocity in the kinetic theory the boltzmann equation can model richer and physically more realistic boundary effects more so than the fluid equations such as the navier stokes this motivates our study the reflection of fluid waves off the boundary is indicated in the figure for the case only one of the euler wave velocities is positive

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

the behavior of flux difference splitting schemes near slowly moving shock waves abstract an investigation of the behavior of shock capturing schemes which compute the numerical flux from a solution of riemann s problem is performed the schemes of godunov roe and osher are examined for a one dimensional model problem consisting of a nearly stationary shock both scalar and systems of equations are examined it is found that for slow shocks there is a significant error generated when solving systems of equations while the scalar results are well behaved this error consists of a long wavelength noise in the downstream running wave families that is not effectively damped by the dissipation of the scheme the source of this error is shown and the implications for the performance of these schemes are considered this error may contribute to the slow convergence to steady state reported by many researchers

on algebras obtained by tensor product abstract let p be a quadratic operad with only one generating operation we define an associated maximal operad p such that for any p algebra a and p algebra b the algebra a b is again a p algebra for the classical tensor product

pointwise error estimates for scalar conservation laws with piecewise smooth solutions we introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation and in fact more general approximations to scalar conservation laws with piecewise smooth solutions to this end we derive a transport inequality for an appropriately weighted error function the key ingredient in our approach is a one sided interpolation inequality between classical l1 error estimates and lip stability bounds the one sided interpolation interesting for its own sake enables us to convert a global l1 result into a nonoptimal local estimate this in turn provides the necessary bounds on the coefficients of the above mentioned transport inequality estimates on the weighted error then follow from the maximum principle and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation unlike previous works in this direction our method can deal with finitely many waves with possible collisions moreover in our approach one does not follow the characteristics but instead makes use of the energy method and hence this approach could be extended to other types of approximate solutions

stability of periodic solutions of conservation laws with viscosity pointwise bounds on the green function we investigate stability of periodic traveling wave solutions of systems of conservation laws with viscosity within the abstract evans function framework established by r a gardner our main result generalizing the work of zumbrun and howard in the traveling front or pulse setting is to establish sharp pointwise bounds on the green function for the linearized evolution equations provided that an appropriate evans function condition applies to the linearized operator about the wave this condition is equivalent to a spectral stability criterion introduced by schneider in the context of periodic reaction diffusion waves an immediate consequence is that strong spectral stability in the sense of schneider implies linearized l1 l p asymptotic stability for all p 1 on the other hand we show that the strict version of schneider s condition generically fails in the conservation law setting leading to complicated new metastable behavior reminiscent of that seen for degenerate neutrally stable families in the traveling front or pulse case our results apply also to the reaction diffusion setting sharpening at the linearized level results obtained by schneider using weighted norm and bloch decomposition methods

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

on the regularity of approximate solutions to conservation laws with piecewise smooth solutions in this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted w1 1 space convergence rate for the derivative of the approximate solutions is established under the assumption that a weak pointwise error estimate is given in other words we are able to convert weak pointwise error estimates to optimal error bounds in a weighted w1 1 space for convex conservation laws the assumption of a weak pointwise error estimate is verified by tadmor siam j numer anal 28 1991 pp 891 906 therefore one immediate application of our w1 1 convergence theory is that for convex conservation laws we indeed have w1 1 error bounds for the approximate solutions to conservation laws furthermore the co en pointwise error estimates of tadmor and tang siam j numer anal 36 1999 pp 1739 1758 are recovered by the use of the w1 1 convergence result

stability of one dimensional boundary layers by using green s functions the aim of this paper is to investigate the stability of one dimensional boundary layers of parabolic systems as the viscosity goes to 0 in the noncharacteristic case and more precisely to prove that spectral stability implies linear and nonlinear stability of approximate solutions in particular we replace the smallness condition obtained by the energy method 10 13 by a weaker spectral condition 2001 john wiley sons inc

initial and shock layers for boltzmann equation we study an initial value problem of the boltzmann equation with a euler shock wave as initial data our analysis exhibits the presence of three singular layers the initial layer formation layer and the shock layer an approximate solution is constructed based on a solution of the burgers type equation for the formation layer time scale the macroscopic conservation laws are preserved for the approximate solution the green s function of the linearized boltzmann equation around the approximate solution is constructed by a modification of the mathbb t mathbb c scheme introduced by yu nonlinear wave propagation over a boltzmann shock profile 2013 with the green s function approach and a wave tracing method one shows that the error of the approximate solution converges to zero with the convergent rate o 1 log varepsilon varepsilon 1 t 1 sigma 0 in pointwise norm cdot l infty xi 3 around the shock profile for sigma 0 in 0 frac 1 2 and with the rate o 1 varepsilon 1 t 1 2 outside the shock zone where varepsilon is the strength of the weak hyperbolic shock wave

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

a new entropy functional for a scalar conservation law in this paper we introduce a new entropy functional for a scalar convex conservation law that generalizes the traditional concept of entropy of the second law of thermodynamics the generalization has two aspects the new entropy functional is defined not for one but for two solutions it is defined in terms of the l1 distance between the two solutions as well as the variations of each separate solution in addition it is decreasing in time even when the solutions contain no shocks and is therefore stronger than the traditional entropy even in the case when one of the solutions is zero 1999 john wiley sons inc

pointwise convergence to shock waves for viscous conservation laws we are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws it is shown that besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and burgers equations a perturbation also gives rise to algebraically decaying terms which measure the coupling of waves of different characteristic families our technique is a combination of time asymptotic expansion construction of approximate green functions and analysis of nonlinear wave interactions the pointwise estimates yield optimal lp convergence of the perturbation to the shock and diffusion waves 1 p the new approach of obtaining pointwise estimates based on the green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks these are being explored elsewhere 1997 john wiley sons inc

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

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

dimension theorem for free ternary partially associative algebras and applications abstract an n ary algebra is a vector space provided with a multiplication given by an n linear map a 3 ary algebra is said to be partially associative if the product x y z satisfies x 1 x 2 x 3 x 4 x 5 x 1 x 2 x 3 x 4 x 5 x 1 x 2 x 3 x 4 x 5 0 we compute the dimensions of the homogeneous components of degree 7 in the free 3 ary partially associative algebra over a field of characteristic zero and give a system of generators for all the homogeneous components we show that this system is a basis if and only if none of these generators is zero as a consequence we show that the corresponding operad doesn t verify the koszul property we also prove that any 3 ary partially associative algebra is solvable of derived length at most 2 and any 3 ary partially associative commutative algebra is nilpotent of index at most 4 for arbitrary integer n we describe the operadic cohomology of the n ary partially associative algebras

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

on the method of modified equations i asymptotic analysis of the euler forward difference method the method of modified equations is studied as a technique for the analysis of finite difference equations the non uniqueness of the modified equation of a difference method is stressed and three kinds of modified equations are introduced the first modified or equivalent equation is the natural pseudo differential operator associated to the original numerical method linear and nonlinear combinations of the equivalent equation and their derivatives yield the second modified or second equivalent equation and the third modified or simply modified equation respectively for linear problems with constant coefficients the three kinds of modified equations are equivalent among them and to the original difference scheme for nonlinear problems the three kinds of modified equations are asymptotically equivalent in the sense that an asymptotic analysis of these equations with the time step as small parameter yields exactly the same results in this paper both regular and multiple scales asymptotic techniques are used for the analysis of the euler forward difference method and the resulting asymptotic expansions are verified for several nonlinear autonomous ordinary differential equations it is shown that when the resulting asymptotic expansion is uniformly valid the asymptotic method yields very accurate results if the solution of the leading order equation is smooth and does not blow up even for large step sizes

elimination of first order errors in time dependent shock calculations first order errors downstream of shocks have been detected in computations with higher order shock capturing schemes in one and two dimensions we use matched asymptotic expansions to analyze the phenomenon for one dimensional time dependent hyperbolic systems and show how to design the artificial viscosity term in order to avoid the first order error numerical computations verify that second order accurate solutions are obtained

existence and asymptotic stability of relaxation discrete shock profiles kin this paper we study the asymptotic nonlinear stability of discrete shocks of the relaxing scheme for approximating the general system of nonlinear hyperbolic conservation laws the existence of discrete shocks is established by suitable manifold construction and it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in l 2 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 relaxing scheme for the single shock solution of the system of hyperbolic conservation laws these results are proved by using both a weighted norm estimate and a characteristic energy method based on the internal structures of the discrete shocks

compressible navier stokes equations with zero heat conductivity abstract we study the large time behavior of solutions of the compressible navier stokes equations without heat conductivity acoustic waves are dissipative due to the viscosity entropy waves are nondissipative due to lack of heat conductivity thus the system is of composite type the simplest prototype of this class of nonlinear partial differential equations we study the nonlinear interactions of waves using a combination of energy method weighted energy method and pointwise estimates

invariant manifolds for steady boltzmann flows and applications we develop a theory of invariant manifolds for the steady boltzmann equation and apply it to the study of boundary layers and nonlinear waves the steady boltzmann equation is an infinite dimensional differential equation so the standard center manifold theory for differential equations based on spectral information does not apply here instead we employ a time asymptotic approach using the pointwise information of green s function for the construction of the linear invariant manifolds at the resonance cases when the mach number at the far field is around one of the critical values of 1 0 or 1 the truly nonlinear theory arises in such a case there are wave patterns combining the fast decaying knudsen type and slow varying fluid like waves the key knudsen manifolds consisting of only knudsentype layers are constructed through delicate analysis of identifying the singular behavior around the critical mach numbers around mach number 1 the fluidlike waves are compressive and expansive waves and around the mach number 0 they are linear thermal layers the quantitative analysis of the fluid like waves is done using the reduction of dimensions to the center manifolds two scale nonlinear dynamics based on those on the knudsen and center manifolds are formulated for the study of the global dynamics of the combined wave patterns there are striking bifurcations in the transition of evaporation to condensation and in the transition of the milne s problem with a subsonic far field to one with a supersonic far field the analysis of these wave patterns allows us to understand the sone diagram for the study of the complete condensation boundary value problem the monotonicity of the boltzmann shock profiles a problem that initially motivated the present study is shown as a consequence of the quantitative analysis of the nonlinear fluid like waves

q state potts models in d dimensions migdal kadanoff approximation the first and second order phase transitions of the q state potts models are obtained in arbitrary dimension d critical and tricritical behaviours merge and annihilate at qc d clearing the way to first order transitions at q qc d by the condensation of effective vacancies the value of qc d decreases with increasing d from diverging as exp 2 d 1 at d to 1 to qc 2 3 81 cf exact value of 4 to lower values at d 2 for given d a changeover in critical behaviour occurs at q1 d as the critical fixed points merge from the potts lattice gas region to the undiluted potts limit it is suggested that the power law singularities of the percolation problem q to 1 have logarithmic corrections

unstable godunov discrete profiles for steady shock waves given a system of n conservation laws ut f u x 0 the steady shock waves when processed by the godunov scheme admit rather simple discrete profiles one shows that the linear stability of these profiles depends only on the location of the eigenvalues of some endomorphism of an n 1 dimensional space applying our theory to the gas dynamics with the perfect gas law p ga 1 rho e we construct unstable profiles for values of ga between 1 and ga and rather strong shocks since ga in 7 5 5 3 our result applies to the air but not to monoatomic gases finally we illustrate this analysis by some numerical experiments both with ga ga

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