traveling wave solutions of convection diffusion systems by center manifold reduction a convection di usion system in one space dimension is a partial di erential equation of the form ut a u ux b u ux x 1 1 here u rn and a u and b u are n n matrices that we shall assume are c2 functions of u if we ignore di usion we have the convection system ut a u ux 0 1 2 if a u df u for some 4ux function f rn rn then eq 1 2 becomes ut f u x 0 1 3 a system of conservation laws while eq 1 1 becomes ut f u x b u ux x 1 4 a system of viscous conservation laws the conservation law case occurs more often in applications and is far better studied see 5 and 12 however equations of the

spectral stability of small amplitude viscous shock waves in several space dimensions a planar viscous shock profile of a hyperbolic parabolic system of conservation laws is a steady solution in a moving coordinate frame the asymptotic stability of viscous profiles and the related vanishing viscosity limit are delicate questions already in the well understood case of one space dimension and even more so in the case of several space dimensions it is a natural idea to study the stability of viscous profiles by analyzing the spectrum of the linearization about the profile the evans function method provides a geometric dynamical systems framework to study the eigenvalue problem in this approach eigenvalues correspond to zeros of an essentially analytic function mathcal e rho lambda rho omega which detects nontrivial intersections of the so called stable and unstable spaces that is spaces of solutions that decay on one or the other side of the shock wave respectively in a series of pioneering papers kevin zumbrun and collaborators have established in various contexts that spectral stability that is the non vanishing of mathcal e rho lambda rho omega and the non vanishing of the lopatinski kreiss majda function imply nonlinear stability of viscous shock profiles in several space dimensions in this paper we show that these conditions hold true for small amplitude extreme shocks under natural assumptions this is done by exploiting the slow fast nature of the small amplitude limit which was used in a previous paper by the authors to prove spectral stability of small amplitude shock waves in one space dimension geometric singular perturbation methods are applied to decompose the stable and unstable spaces into subbundles with good control over their limiting behavior three qualitatively different regimes are distinguished that relate the small strength epsilon of the shock wave to appropriate ranges of values of the spectral parameters various rescalings are used to overcome apparent degeneracies in the problem caused by loss of hyperbolicity or lack of transversality

vanishing viscosity solutions of nonlinear hyperbolic systems we consider the cauchy problem for a strictly hyperbolic n n system in one space dimension ut a u ux 0 assuming that the initial data have small total variation we show that the solutions of the viscous approximations ut a u ux euxx are defined globally in time and satisfy uniform bv estimates independent of e moreover they depend continuously on the initial data in the l 1 distance with a lipschitz constant independent of t e letting e 0 these viscous solutions converge to a unique limit depending lipschitz continuously on the initial data in the conservative case where a df is the jacobian

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

strong viscous shocks for systems of conservation laws with a high order of dissipation on etudie l existence de solutions onde de propagation pour le systeme f u u t g u u x 1 k 1 d x k ad x k u ou la matrice a est une fonction c k de ses variables

nonexistence of riemann solutions and majda pego instability abstract we study riemann problems for the shallow water equations we consider weak self similar riemann solutions consisting of constant states rarefaction waves and or jump discontinuities that satisfy the viscous profile entropy criterion with a positive definite symmetric viscosity matrix we prove that for a generic symmetric positive definite viscosity matrix there is an open set of riemann initial data for which a weak self similar riemann solution does not exist we show that this happens for the hyperbolic initial data that is unstable in the sense studied by majda and pego we prove that such initial data always exist for positive definite symmetric nondiagonal viscosity matrices in the work that follows previous work by the authors in press nonlinear anal we show that in the situations presented in this paper measure value solutions exhibiting continuously generated oscillations take place the results of the present paper provide a new insight into the role of the viscous profile entropy criterion and the majda pego instability in the existence of riemann solutions for nonlinear conservation laws

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

logarithmic voltage anomalies in random resistor networks the authors investigate the behaviour of the maximum voltage drop across the bonds in a random resistor network above the percolation threshold on the basis of numerical simulations on randomly diluted l l square lattice networks we find that the average value of the maximum voltage vmax p l exhibits a peak as a function of the bond concentration p which is located above the percolation threshold this peak value appears to grow logarithmically with l while the location of the peak appears to approach the percolation threshold very slowly as l increases to help understand these results they introduce the bubble model a quasi one dimensional structure in which the system length varies exponentially in the system width this model is exactly soluble and a rather good description of percolation in systems of greater than one dimension can be obtained moreover within the bubble model the peak value of the maximum voltage increases as in l while the peak location approaches the percolation threshold as in in l in l in good agreement with our numerical results

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

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

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

time implicit high order discontinuous galerkin method with reduced evaluation cost an efficient and robust time integration procedure for a high order discontinuous galerkin method is introduced for solving nonlinear second order partial differential equations the time discretization is based on an explicit formulation for the hyperbolic term and an implicit formulation for the parabolic term the procedure uses an iterative algorithm with reduced evaluation cost the size of the linear system to be solved is greatly reduced thanks to partial uncoupling in space between low order and high order degrees of freedom numerical examples are presented for the nonlinear convection diffusion equation in one and two dimensions including steady and unsteady flow problems the performance of the present method is investigated in terms of cpu time and compared to a fully implicit method a von neumann stability analysis is carried out in order to determine the stability and damping properties of the method besides a fairly reduced cpu effort numerical results demonstrate better convergence properties of the present algorithm when compared to the fully implicit method

on the riemann problem for non conservative hyperbolic systems we consider the construction and the properties of the riemann solver for the hyperbolic system

dynamical stability of phase transitions in the p system with viscosity capillarity we show for the p system with viscosity capillarity that all heteroclinic phase transitions lying sufficiently close to the maxwell line are linearly dynamically stable under localized perturbations independent of the form of the flux on the other hand we show that nearby homoclinic phase transitions involving two or more changes in phase are exponentially unstable these results together imply that stationary and slowly moving phase transitions are dynamically stable if and only if they are monotone in analogy with results obtained by carr gurtin and slemrod in the variational setting arch rational mech anal 86 1984 pp 317 351 this research is significant as the first proof of stability of a strong viscous shock wave for systems our analysis uses the technical framework developed in r gardner and k zumbrun comm pure appl math 51 1998 pp 797 855 k zumbrun and p howard indiana univ math j 47 1998 pp 741 871 together with a nonstandard energy estimate

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

leonard triples from the equitable basis of sl2 abstract we construct leonard triples on the irreducible finite dimensional sl 2 modules using the equitable basis for sl 2 we show that the leonard triples are of either racah or krawtchouk type

remarks on the stability of shock profiles for conservation laws with dissipation two remarks are made the first is to establish the stability of monotone shock profiles of the kdv burgers equation based on an energy method of goodman the second remark illustrates specifically in burgers equation that uniform rates of decay are not to be expected for perturbations of shock profiles in typical norms introduction we will prove two results the first is to establish the orbital stability of monotone shock profile solutions 4 x st of the kdv burgers equation 1 1 ut uux uxx uxxx 0 0 the second is to show that for shock profile solutions of viscous conservation laws and specifically for burgers equation 1 2 ut uux uxx one cannot expect perturbations to decay at any uniform rate algebraic or otherwise in ordinary norms such as lp the first result is related to goodman s conditional stability result in 5 for small amplitude shock profiles x st in systems of genuinely nonlinear conservation laws with viscosity u f u x buxx u e rm the second is intended to be relevant to the question of full stability for shock profiles of such systems the nonlinear stability to small perturbations of constant state solutions u0 x c of the system u f u x b u ux x where b c is a strictly stable viscosity matrix see 9 can be demonstrated based on a uniform rate of decay of solutions to the linearized equation the method used is similar to that developed in 8 and 10 for systems in n 1 space dimensions faster decay rates in higher space dimensions make the method easier to apply when n is large but it does just work for conservation laws with stable viscosities in one space dimension see 13 thus for example assuming c 0 if iiuo iih6 ll is sufficiently small then iiu t llh4 ct 1 4iiuoii h4 n la in this note we want to point out that such uniform decay rates cannot be valid for perturbations of shock profiles in particular for the shock profile solution of burgers equations we construct a family of perturbations which decay to zero but do so at no uniform rate in most ordinary norms including all lp norms thus for received by the editors october 15 1984 and in revised form december 19 1984 1980 mathematics subject classification primary 35q20 35l65 1985 american mathematical society 0002 9947 85 1 00 25 per page

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

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

boundary layers in weak solutions of hyperbolic conservation laws this paper is concerned with the initial boundary value problem for a nonlinear hyperbolic system of conservation laws we study the boundary layers that may arise in approximations of entropy discontinuous solutions we consider both the vanishing viscosity method and finite difference schemes lax friedrichs type schemes and the godunov scheme we demonstrate that different regularization methods generate different boundary layers hence the boundary condition can be formulated only if an approximation scheme is selected first assuming solely uniform l infty bounds on the approximate solutions and so dealing with l infty solutions we derive several entropy inequalities satisfied by the boundary layer in each case under consideration a young measure is introduced to describe the boundary trace when a uniform bound on the total variation is available the boundary young measure reduces to a dirac mass from the above analysis we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not each formulation is based on a set of admissible boundary values following the terminology of dubois lefloch 15 the local structure of these sets and the well posedness of the corresponding initial boundary value problem are investigated the results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics

center manifolds for a class of degenerate evolution equations and existence of small amplitude kinetic shocks abstract we construct center manifolds for a class of degenerate evolution equations including the steady boltzmann equation and related kinetic models establishing in the process existence and behavior of small amplitude kinetic shock and boundary layers notably for boltzmann s equation we show that elements of the center manifold decay in velocity at near maxwellian rate in accord with the formal chapman enskog picture of near equilibrium flow as evolution along the manifold of maxwellian states or grad moment approximation via hermite polynomials in velocity our analysis is from a classical dynamical systems point of view with a number of interesting modifications to accommodate ill posedness of the underlying evolution equation

symmetrization of dissipative dispersive traveling waves for systems of conservation laws abstract this work addresses existence of traveling waves for systems of conservation laws through an hamiltonian formalism a related hamilton jacobi equation is given and studied our proofs rely on the construction of a cost functional and require a smallness condition for the dispersion with respect to diffusion some numerical applications are also presented concerning a model phase transition problem

wave interactions and variation estimates for self similar zero viscosity limits in systems of conservation laws we consider the problem of self similar zero viscosity limits for systems ofn conservation laws first we give general conditions so that the resulting boundary value problem admits solutions the obtained existence theory covers a large class of systems in particular the class of symmetric hyperbolic systems second we show that if the system is strictly hyperbolic and the riemann data are sufficiently close then the resulting family of solutions is of uniformly bounded variation and oscillation third we construct solutions of the riemann problem via self similar zero viscosity limits and study the structure of the emerging solution and the relation of self similar zero viscosity limits and shock profiles the emerging solution consists ofn wave fans separated by constant states each wave fan is associated with one of the characteristic fields and consists of a rarefaction a shock or an alternating sequence of shocks and rarefactions so that each shock adjacent to a rarefaction on one side is a contact discontinuity on that side at shocks the solutions of the self similar zero viscosity problem have the internal structure of a traveling wave

nonuniqueness for the vanishing viscosity solution with fixed initial condition in a nonstrictly hyperbolic system of conservation laws we consider the riemann problem for a system of two decoupled nonstrictly hyperbolic burgers like conservation equations with added artificial viscosity we analytically establish two different vanishing viscosity limits for the solution of this system which correspond to the two cases where one of the viscosities vanishes much faster than the other this is done without altering the initial condition as is necessary with travelling wave methods numerical evidence is then provided to show that when the two viscosities vanish at the same rate the solution converges to a limit that lies strictly between the two previously established limits finally we use control theory to explain the mechanism behind this nonuniqueness behavior which indicates other systems of nonstrictly hyperbolic conservation laws where nonuniqueness will occur

boundary layers for compressible navier stokes equations with outflow boundary condition abstract in this paper we study the asymptotic relation between the solutions to the initial boundary value problem of the one dimensional compressible full navier stokes equations with outflow boundary condition and the associated euler equations we assume all the three characteristics to the corresponding euler equations are all negative up to some small time then we prove the existence and the stability of the boundary layers as long as the strength of the boundary layers is suitably small

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

leonard triples from leonard pairs constructed from the standard basis of the lie algebra sl2 let k denote an algebraically closed field of characteristic zero and d 3 denote an integer an ordered pair of matrices a a is a leonard pair on the vector space kd 1 if we can find invertible matrices s1 and s2 md 1 k such that i s1 1as1 is diagonal and s1 1a s1 is irreducible tridiagonal and ii s2 1a s2 is diagonal and s2 1as2 is irreducible tridiagonal we extend this concept to three matrices we say an ordered triple of matrices a a and a is a leonard triple on kd 1 if we can find invertible matrices s1 s2 and s3 such that i s1 1as1 is diagonal and both s1 1a s1 and s1 1a s1 are irreducible tridiagonal ii s2 1a s2 is diagonal and both s2 1as2 and s2 1a s2 are irreducible tridiagonal and iii s3 1a s3 is diagonal and both s3 1as3 and s3 1a s3 are irreducible tridiagonal let a 0d010d 120 10d0anda diag d d 2 d be d 1 d 1 matrices then a a is a leonard pair on kd 1 we determine all the matrices a such that a a a form a leonard triple on kd 1

traveling waves solution of convection diffusion systems whose convection terms are weakly nonconservative application to the modeling of two phase fluid flows the paper is concerned with the derivation of explicit approximate jump conditions for the shock waves solutions of hyperbolic systems in nonconservation form extracted from a known second order convection diffusion system the shock waves are defined as the limit as the diffusion tends to zero of the traveling waves solution of the second order system although no explicit jump conditions may be found for general hyperbolic systems in nonconservation form we derive an asymptotic expansion of the nonexplicit jump relations in the case of two phase fluid flows we indeed observe that the terms in nonconservation form arise from pressure gradient forces which act on droplets and are small in the sense that the a small number appears in front of them we obtain an asymptotic expansion of the nonexplicit jump relations whose first two terms can be computed explicitly

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

existence and stability of noncharacteristic boundary layers for the compressible navier stokes and viscous mhd equations for a general class of hyperbolic parabolic systems including the compressible navier stokes and compressible mhd equations we prove existence and stability of noncharacteristic viscous boundary layers for a variety of boundary conditions including classical navier stokes boundary conditions our first main result using the abstract framework established by the authors in the companion work gues et al in j differ equ 244 309 387 2008 is to show that existence and stability of arbitrary amplitude exact boundary layer solutions follow from a uniform spectral stability condition on layer profiles that is expressible in terms of an evans function uniform evans stability whenever this condition holds we give a rigorous description of the small viscosity limit as the solution of a hyperbolic problem with residual boundary conditions our second is to show that uniform evans stability for small amplitude layers is equivalent to evans stability of the limiting constant layer which in turn can be checked by a linear algebraic computation finally for a class of symmetric dissipative systems including the physical examples mentioned above we carry out energy estimates showing that constant and thus small amplitude layers always satisfy uniform evans stability this yields existence of small amplitude multi dimensional boundary layers for the compressible navier stokes and mhd equations for both equations these appear to be the first such results in the compressible case

spectral stability of ideal gas shock layers extending recent results in the isentropic case we use a combination of asymptotic ode estimates and numerical evans function computations to examine the spectral stability of shock wave solutions of the compressible navier stokes equations with ideal gas equation of state our main results are that in appropriately rescaled coordinates the evans function associated with the linearized operator about the wave i converges in the large amplitude limit to the evans function for a limiting shock profile of the same equations for which internal energy vanishes at one end state and ii has no unstable positive real part zeros outside a uniform ball thus the rescaled eigenvalue ode for the set of all shock waves augmented with the nonphysical limiting case form a compact family of boundary value problems that can be conveniently investigated numerically an extensive numerical evans function study yields one dimensional spectral stability independent of amplitude for gas constant in 1 2 3 and ratio of heat conduction to viscosity coefficient within 0 2 5 1 4 1 47 for air other values may be treated similarly but were not considered the method of analysis extends also to the multi dimensional case a direction that we shall pursue in a future work

existence of traveling wave solutions for hyperbolic systems of balance laws this paper is concerned with traveling wave solutions for hyperbolic systems of balance laws satisfying a stability condition and a kawashima like condition we are interested in the case where the traveling wave equations have a singularity which is absent for 2 2 systems satisfying the two conditions to deal with the singularity we reduce the problem to a parametrized one without singularity by using the center manifold theorem for the parametrized problem we prove the existence of solutions by modifying an existing argument in the literature in this way we show the existence of traveling wave solutions

a numerical study of a rotationally degenerate hyperbolic system part i the riemann problem abstract we study numerically the riemann problem for a 2 x 2 system of conservation laws with a cubic flux function a particular case of the class of models introduced by keyfitz and kranzer the system is not strictly hyperbolic and the classical lax theory for hyperbolic systems is not directly applicable correspondingly some numerical schemes which are accurate for strictly hyperbolic systems are not well behaved for this example when they do work different schemes yield markedly different results for certain data we explain this effect by observing that near these data viscous regularization is non uniform as the viscosity tends to zero this fact does not contradict the well posedness of the hyperbolic model it does imply that precise control of the viscosity introduced into a computational method is crucial for generating the correct numerical solutions we examine all of these issues and comment on their implications for similar systems which arise in continuum mechanics

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

stability of periodic solutions of conservation laws with viscosity analysis of the evans function nonclassical conservation laws with viscosity arising in multiphase fluid and solid mechanics exhibit a rich variety of traveling wave phenomena including homoclinic pulse type and periodic solutions along with the standard heteroclinic shock or front type solutions here we investigate stability of periodic traveling waves within the abstract evans function framework established by r a gardner our main result is to derive a useful stability index analogous to that developed by gardner and zumbrun in the traveling front or pulse context giving necessary conditions for stability with respect to initial perturbations that are periodic on the same period t as the traveling wave moreover we show that the periodic stability index has an interpretation analogous to that of the traveling front or pulse index in terms of well posedness of an associated riemann problem for an inviscid medium now to be interpreted as allowing a wider class of measure valued solutions or alternatively in terms of existence and nonsingularity of a local mass map from perturbation mass to potential time asymptotic t periodic states a closely related calculation yields also a complementary long wave stability criterion necessary for stability with respect to periodic perturbations of arbitrarily large period nt n we augment these analytical results with numerical investigations analogous to those carried out by brin in the traveling front or pulse case approximating the spectrum of the linearized operator about the wave

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

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

stability of 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

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

temporal development of invasion percolation the authors consider the dynamics of invasion percolation for different models this process exhibits scaling behaviours which in the simplest case can be related to some critical static properties of usual percolation in particular the concept of bursts is shown to be of fundamental importance the critical exponents they derive are checked numerically on a case of invasion percolation where growth is restricted to the hull of the existing cluster

spectral stability of shock waves associated with not genuinely nonlinear modes we study viscous shock waves that are associated with a simple mode r of a system ut f u x uxx of conservation laws and that connect states on either side of an inflection hypersurface in state space at whose points r 0 and r 2 0 such loss of genuine nonlinearity the simplest example of which is the cubic scalar conservation law ut u3 x uxx occurs in many physical systems we show that such shock waves are spectrally stable if their amplitude is sufficiently small the proof is based on a direct analysis of the eigenvalue problem by means of geometric singular perturbation theory well chosen rescalings are crucial for resolving degeneracies by results of zumbrun the spectral stability shown here implies nonlinear stability of these shock waves

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

finite energy method for compressible fluids the navier stokes korteweg model this is the first of a series of papers devoted to the initial value problem for the one dimensional euler system of compressible fluids and augmented versions containing higher order terms in the present paper we are interested in dispersive shock waves and analyze the zero viscosity capillarity limit associated with the navier stokes korteweg system specifically we establish the existence of finite energy solutions as well as their convergence toward entropy solutions to the euler system our method of proof combines energy and effective energy estimates a nonlinear sobolev inequality high integrability properties for the mass density and for the velocity and compactness properties based on entropies 2015 wiley periodicals inc

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

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

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

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

singular limits in phase dynamics with physical viscosity and capillarity following pioneering work by fan and slemrod who studied the e ect of artificial viscosity terms we consider the sys tem of conservation laws arising in liquid vapor phase dynamics with physical viscosity and capillarity e ects taken into account following dafermos we consider self similar solutions to the rie mann problem and establish uniform total variation bounds al lowing us to deduce new existence results our analysis cover both the hyperbolic and the hyperbolic elliptic regimes and apply to arbitrarily large riemann data the proofs rely on a new technique of reduction to two coupled scalar equations associated with the two wave fans of the system strong l 1 convergence to a weak solution of bounded variation is established in the hyperbolic regime while in the hyperbolic elliptic regime a stationary singularity near the axis separating the two wave fans or more generally an almost stationary oscil lating wave pattern of thickness depending upon the capillarity viscosity ratio are observed which prevent the solution to have globally bounded variation

totally bipartite leonard pairs and totally bipartite leonard triples of q racah type abstract let k denote an algebraically closed field of characteristic zero let v denote a vector space over k with finite positive dimension a leonard pair on v is an ordered pair of linear transformations in end v such that for each of these transformations there exists a basis for v with respect to which the matrix representing that transformation is diagonal and the matrix representing the other transformation is irreducible tridiagonal whenever these tridiagonal matrices are bipartite the leonard pair is said to be totally bipartite the notion of a leonard triple and the corresponding notion of totally bipartite are similarly defined in this paper we first classify up to isomorphism the totally bipartite leonard pairs the classification reveals that these leonard pairs are of the q racah krawtchouk or bannai ito type then we determine all leonard triples extended from given totally bipartite leonard pairs of q racah type finally we classify up to isomorphism the totally bipartite leonard triples of q racah type

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

leonard triples of q racah type and their pseudo intertwiners abstract let f denote a field and let v denote a vector space over f with finite positive dimension pick a nonzero q f such that q 4 1 and let a b c denote a leonard triple on v that has q racah type we show that there exist invertible w w w in end v such that i a commutes with w and w 1 b w c ii b commutes with w and w 1 c w a iii c commutes with w and w 1 a w b moreover each of w w w is unique up to multiplication by a nonzero scalar in f we show that the three elements w w w w w w mutually commute and their product is a scalar multiple of the identity a number of related results are obtained we call w w w the pseudo intertwiners for a b c

on the shock wave spectrum for isentropic gas dynamics with capillarity abstract we consider the stability problem for shock layers in slemrod s model of an isentropic gas with capillarity we show that these traveling waves are monotone in the weak capillarity case and become highly oscillatory as the capillarity strength increases using a spectral energy estimate we prove that small amplitude monotone shocks are spectrally stable we also show that monotone shocks have no unstable real spectrum regardless of amplitude this implies that any instabilities of these monotone traveling waves if they exist must occur through a hopf like bifurcation where one or more conjugate pairs of eigenvalues cross the imaginary axis we then conduct a systematic numerical evans function study which shows that monotone and mildly oscillatory profiles in an adiabatic gas are spectrally stable for moderate values of shock and capillarity strengths in particular we show that the transition from monotone to nonmonotone profiles does not appear to trigger any instabilities

stable viscosities and shock profiles for systems of conservation laws abstract many equations of mathematical physics take the form of nonlinear hyperbolic systems of conservation laws with small dissipative effects neglected typically smooth solutions must develop discontinuities shocks in finite time re incorporating dissipation helps select those discontinuities which are physically relevant for this purpose many different sorts of dissipation will do in particular the physical viscosity is typically degenerate and not convenient in this paper the author provide an understanding of what high order viscosity terms smooth the physical discontinuities a natural class of admissible viscosity terms is determined based on a simple linearized stability criterion in addition they determine a class of degenerate second order viscosity terms of physical type which are admissible these results are applied to the equations of compressible fluid dynamics to determine what conditions ensure the existence of the shock layer with viscosity and heat conduction this should be of interest to others interested in general equations of state for compressible fluids such as those investigating phase transitions

leonard triples extended from a given totally almost bipartite leonard pair of bannai ito type let kk denote a field of characteristic zero and let d denote an integer at least 3 let a 1 d 02d 10102d202d 13 d 3d 10d 20dd 1 and a diag 1 d 2d 1 7 5 3 1 a diag 1 d 2d 1 7 5 3 1 be two matrices in matd 1 k matd 1 k then a a a a is a totally almost bipartite leonard pair on kd 1kd 1 of bannai ito type in this paper we determine all the matrices ae matd 1 k ae matd 1 k such that a a aea a ae form a leonard triple on kd 1kd 1

one dimensional stability of parallel shock layers in isentropic magnetohydrodynamics abstract extending investigations of barker humpherys lafitte rudd and zumbrun for compressible gas dynamics and freistuhler and trakhinin for compressible magnetohydrodynamics we study by a combination of asymptotic ode estimates and numerical evans function computations the one dimensional stability of parallel isentropic magnetohydrodynamic shock layers over the full range of physical parameters shock amplitude strength of imposed magnetic field viscosity magnetic permeability and electrical resistivity for a law gas with 1 3 other values may be treated similarly but were not checked numerically depending on magnetic field strength these shocks may be of fast lax intermediate overcompressive or slow lax type however the shock layer is independent of magnetic field consisting of a purely gas dynamical profile in each case our results indicate stability interesting features of the analysis are the need to renormalize the evans function in order to pass continuously across parameter values where the shock changes type or toward the large amplitude limit at frequency 0 and the systematic use of winding number computations on riemann surfaces

conductance and resistance jumps in finite size random resistor networks when bonds are removed one by one and at random from a finite size resistor network the conductance or resistance does not change continuously but rather a sequence of conductance resistance jumps for various sizes occurs the larger jumps arise from those bonds which carry a relatively large current just before they are cut the authors report on numerical simulations of these jumps of a random resistor network on the square lattice they also give a scaling argument to account for this phenomenon which yields the number of conductance jumps larger than delta g scaling as delta g lambda g with lambda g d nu d nu t where t is the conductivity exponent and nu is the percolation correlation length exponent equivalently the number of resistance jumps greater than delta r scales as delta r lambda r with lambda r d nu d nu t these predictions account for the data on a qualitative level only however and they discuss some possible mechanisms for the quantitative discrepancies

stability of undercompressive viscous shock profiles of hyperbolic parabolic systems abstract extending to systems of hyperbolic parabolic conservation laws results of howard and zumbrun for strictly parabolic systems we show for viscous shock profiles of arbitrary amplitude and type that necessary spectral evans function conditions for linearized stability established by mascia and zumbrun are also sufficient for linearized and nonlinear phase asymptotic stability yielding detailed pointwise estimates and sharp rates of convergence in l p 1 p

dual family viscous shock waves in n conservation laws with application to multi phase flow in porous media we consider shock waves satisfying the viscous profile criterion in general systems of n conservation laws we study si j dual family shock waves which are associated with a pair of characteristic families i and j we explicitly introduce defining equations relating states and speeds of si j shocks which include the rankine hugoniot conditions and additional equations resulting from the viscous profile requirement we then develop a constructive method for finding the general local solution of the defining equations for such shocks and derive formulae for the sensitivity analysis of si j shocks under change of problem parameters all possible structures of solutions to the riemann problems containing si j shocks and classical waves are described as a physical application all types of si j shocks with i j are detected and studied in a family of models for multi phase flow in porous media

pointwise green function bounds and stability of combustion waves abstract generalizing similar results for viscous shock and relaxation waves we establish sharp pointwise green function bounds and linearized and nonlinear stability for traveling wave solutions of an abstract viscous combustion model including both majda s model and the full reacting compressible navier stokes equations with artificial viscosity with general multi species reaction and reaction dependent equation of state under the necessary conditions of strong spectral stability i e stable point spectrum of the linearized operator about the wave transversality of the profile as a connection in the traveling wave ode and hyperbolic stability of the associated chapman jouguet square wave approximation notably our results apply to combustion waves of any type weak or strong detonations or deflagrations reducing the study of stability to verification of a readily numerically checkable evans function condition together with spectral results of lyng and zumbrun this gives immediately stability of small amplitude strong detonations in the small heat release i e fluid dynamical limit simplifying and greatly extending previous results obtained by energy methods by liu ying and tesei tan for majda s model and the reactive navier stokes equations respectively

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