fundamental frequency analysis of laminated rectangular plates by differential quadrature method in this paper a differential quadrature method is presented for computation of the fundamental frequency of a thin laminated rectangular plate the partial differential equations of motion for free vibration are solved for the boundary conditions by approximating them by substituting weighted polynomials functions for the differential operator by doing this the coupled partial differential equations of motion are reduced to sets of homogeneous algebraic equations these sets of homogeneous algebraic equations are combined to give a set of general eigenvalue equations for the problem three types of laminated plate problems which include symmetric antisymmetric cross ply and symmetric balanced angle ply laminates are analysed by the method and the results obtained are compared with solutions reported in the literature for other numerical methods the effects of the level of discretization on the accuracy and rate of convergence of the results are also discussed the method presented gives accurate results and is found to use not much computer time

stability of radiative shock profiles for hyperbolic elliptic coupled systems abstract extending previous work with lattanzio and mascia on the scalar in fluid dynamical variables hamer model for a radiative gas we show nonlinear orbital asymptotic stability of small amplitude shock profiles of general systems of coupled hyperbolic elliptic equations of the type modeling a radiative gas that is systems of conservation laws coupled with an elliptic equation for the radiation flux including in particular the standard euler poisson model for a radiating gas the method is based on the derivation of pointwise green function bounds and description of the linearized solution operator with the main difficulty being the construction of the resolvent kernel in the case of an eigenvalue system of equations of degenerate type nonlinear stability then follows in standard fashion through linear estimates derived from these pointwise bounds combined with energy estimates of nonlinear damping type

pointwise estimates and stability for dispersive diffusive shock waves we study the stability and pointwise behavior of perturbed viscous shock waves for a general scalar conservation law with constant diffusion and dispersion along with the usual lax shocks such equations are known to admit undercompressive shocks we unify the treatment of these two cases by introducing a new wave tracking method based on instantaneous projection giving improved estimates even in the lax case another important feature connected with the introduction of dispersion is the treatment of a non sectorial operator an immediate consequence of our pointwise estimates is a simple spectral criterion for stability in all l p norms p 1 for the lax case and p 1 for the undercompressive case our approach extends immediately to the case of certain scalar equations of higher order and would also appear suitable for extension to systems

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

asymptotic behavior near planar transition fronts for the cahn hilliard equation abstract we consider the asymptotic behavior of perturbations of planar wave solutions arising in the cahn hilliard equation in space dimensions d 2 such equations are well known to arise in the study of spinodal decomposition a phenomenon in which rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into regions in which one component or the other is dominant with these regions separated by steep transition layers a critical feature of the cahn hilliard equation in one space dimension 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 the analysis of planar waves in multiple space dimensions is further complicated by the fact that the leading eigenvalue for this linearized operator leading in the case of stability moves into the negative real half plane with cubic scaling 3 where r d 1 denotes a fourier variable associated with spatial components transverse to the planar wave under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from planar wave solutions establishing asymptotic stability for initial perturbations decaying at an appropriate algebraic rate in an l 1 norm of the transverse variables

on the existence of pluricanonical forms on varieties with infinite fundamental group we study the existence of pluricanonical forms on an algebraic variety of general type with infinite fundamental group we have a nonvanishing of mth plurigenus of the variety by that of a general fibre of the so called shafarevich map of the variety

transition to longitudinal instability of detonation waves is generically associated with hopf bifurcation to time periodic galloping solutions we show that transition to longitudinal instability of strong detonation solutions of reactive compressible navier stokes equations is generically associated with hopf bifurcation to nearby time periodic galloping or pulsating solutions in agreement with physical and numerical observation in the process we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated evans function and obtain the first complete nonlinear stability result for strong detonations of the reacting navier stokes equations in the limit as amplitude hence also heat release goes to zero the analysis is by pointwise semigroup techniques introduced by the authors and collaborators in previous works

some mixed hodge structures on l 2 cohomology groups of coverings of kahler manifolds we give methods to compute l 2 cohomology groups of a covering manifold obtained by removing the pullback of a normal crossing divisor to a covering of a compact kahler manifold we prove that in suitable quotient categories these groups admit natural mixed hodge structure whose graded pieces are given by the expected gysin maps

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

nonlinear stability of an undercompressive shock for complex burgers equation though there is strong numerical evidence for the stability of undercompressive shocks their stability has not been verified analytically in particular the energy methods used to analyze stability of standard shocks do not apply here we present the first proof of stability for a particular undercompresive shock the real burgers shock considered as a solution of the complex burgers equation our analysis is by direct calculation of the green s function for the linearized equations combined with pointwise estimates of nonlinear effects a benefit of this method is to obtain fairly detailed information about the solution includingl1 behavior and rates of decay in different regions of space

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

high frequency asymptotics and one dimensional stability of zel dovich von neumann doring detonations in the small heat release and high overdrive limits we establish one dimensional spectral or normal modes stability of zel dovich von neumann doring detonations in the small heat release limit and the related high overdrive limit with heat release and activation energy held fixed verifying numerical observations made by erpenbeck in the 1960s the key technical points are a strategic rescaling of parameters converting the infinite overdrive limit to a finite regular perturbation problem and a careful high frequency analysis depending uniformly on model parameters the latter recovers and extends to arbitrary amplitudes the important result of high frequency stability established by erpenbeck by somewhat different techniques notably the techniques used here yield quantitative estimates well suited for numerical stability investigation

differential quadrature layerwise modeling technique for three dimensional analysis of cross ply laminated plates of various edge supports abstract in this paper a differential quadrature model is proposed which is established within a layerwise framework for the bending analysis of composite laminates three dimensional theory of elasticity is implemented and all physical governing equations are satisfied including characteristics of the continuity and discontinuity of interfacial transverse and in plane displacements strains and stresses along interfaces of the composite laminates thus delivering highly accurate numerical solutions variation in boundary conditions plate aspect and thickness ratios and loading conditions are examined demonstrating the applicability accuracy and stability of the present methodology numerical examples for rectangular cross ply laminated plates subjected to a uniform load sinusoidal load or uniformly distributed moment are studied selected results from these examples are compared with those available in existing published literature

comparison of natural frequencies of laminates by 3 d theory part i rectangular plates abstract the three dimensional theory of laminated plates and shells has been developed by chao et al 10 13 62 63 with many applications to impact and shock modal analyses in this research a complete survey of the literature is made on the free vibration natural frequencies of simply supported rectangular plates various boundary conditions are composed of fixed pin hinge roller and sliding pin supported edges the lowest frequencies are obtained in the present study in comparison with those in earlier studies as a result of the close natural state reached in keeping with the three dimensional boundary and interlaminar continuity conditions via a 3 d augmented energy variational approach

invariants de von neumann des faisceaux analytiques coherents in order to study the group of l 2 holomorphic sections of the pull back to the universal covering space of an holomorphic vector bundle on a compact complex manifold it would be convenient to have a cohomological formalism generalizing atiyah s l 2 index theorem in eys99 such a formalism is proposed in a restricted context to each coherent analytic sheaf cal f on a n dimensionnal smooth projective variety x n and each galois infinite unramified covering pi tilde x to x whose galois group is denoted by gamma l 2 cohomology groups denoted by h q 2 tilde x cal f are attached such that

l1 asymptotic behavior of compressible isentropic viscous 1 d flow we study the large time behavior in l1 of the compressible isentropic viscous 1 d flow under the assumption that the initial data are smooth and small we show that the solutions are approximated by the solutions of a parabolic system and in turn by diffusion waves which are solutions of burgers equations decay rates in l1 are obtained our method is based on the study of pointwise properties in the physical space of the fundamental solution to the linearized system 1994 john wiley sons inc

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

stability of planar shock fronts for multidimensional systems of relaxation equations abstract we investigate stability of multidimensional planar shock profiles of a general hyperbolic relaxation system whose equilibrium model is a system under the necessary assumption of spectral stability and a standard set of structural conditions that are known to hold for many physical systems our main result generalizing the work of kwon and zumbrun in the scalar relaxation case is to establish the bounds on the green s function for the linearized equation and obtain nonlinear l 2 asymptotic behavior sharp decay rate of perturbed weak shock profiles to establish green s function bounds we use the semigroup approach in the low frequency regime and use the energy method for the high frequency bounds separately for the system equilibrium case the analysis of the linearized equation is complicated due to glancing phenomena we treat this difficulty similarly as in the inviscid and viscous systems under the constant multiplicity condition

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

nonlinear stability of time periodic viscous shocks in order to understand the nonlinear stability of many types of time periodic travelling waves on unbounded domains one must overcome two main difficulties the presence of embedded neutral eigenvalues and the time dependence of the associated linear operator this problem is studied in the context of time periodic lax shocks in systems of viscous conservation laws using spatial dynamics and a decomposition into separate floquet eigenmodes it is shown that the linear evolution for the time dependent operator can be represented using a contour integral similar to that of the standard time independent case by decomposing the resulting green s distribution the leading order behavior associated with the embedded eigenvalues is extracted sharp pointwise bounds are then obtained which are used to prove that time periodic lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues the latter conditions hold for example for small oscillation time periodic waves that emerge through a supercritical hopf bifurcation from a family of time independent lax shocks of possibly large amplitude

on the stability of traveling waves in weighted l spaces abstract i present new techniques and results concerning the stability of travelling waves to semilinear parabolic equations i am concerned with those problems in which the essential spectrum of the linearization about the wave contains 0 in these circumstances the stability results of d henry in lecture notes in mathematics vol 840 springer verlag new york 1981 and p bates and c jones dynamics rep 2 1989 1 38 are not applicable as the spectral information does not yield an exponential time decay estimate on the semigroup i show that under certain relatively easily computable conditions the wave is stable in polynomially weighted l spaces furthermore the rate at which the perturbation decays to the wave as t depends on the growth rate of the polynomial near infinity the method of proof uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates for the semigroup are derived through a new technique for estimating the resolvent

conditional stability of unstable viscous shock waves in compressible gas dynamics and mhd extending our previous work in the strictly parabolic case we show that a linearly unstable lax type viscous shock solution of a general quasilinear hyperbolic parabolic system of conservation laws possesses a translation invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small l 1 h 3 perturbations converging time asymptotically to a translate of the unperturbed wave that is for a shock with p unstable eigenvalues we establish conditional stability on a codimension p manifold of initial data with sharp rates of decay in all l p for p 0 we recover the result of unconditional stability obtained by mascia and zumbrun the main new difficulty in the hyperbolic parabolic case is to construct an invariant manifold in the absence of parabolic smoothing

long time stability of multi dimensional noncharacteristic viscous boundary layers we establish long time stability of multi dimensional noncharacteristic boundary layers of a class of hyperbolic parabolic systems including the compressible navier stokes equations with inflow outflow boundary conditions under the assumption of strong spectral or uniform evans stability evans stability has been verified for small amplitude layers by gues metivier williams and zumbrun for large amplitude layers it may be efficiently checked numerically as done in the one dimensional case by costanzino humpherys nguyen and zumbrun

conditional stability of unstable viscous shocks abstract continuing a line of investigation initiated by texier and zumbrun on dynamics of viscous shock and detonation waves we show that a linearly unstable lax type viscous shock solution of a semilinear strictly parabolic system of conservation laws possesses a translation invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small l 1 h 2 perturbations converging time asymptotically to a translate of the unperturbed wave that is for a shock with p unstable eigenvalues we establish conditional stability on a codimension p manifold of initial data with sharp rates of decay in all l p for p 0 we recover the result of unconditional stability obtained by howard mascia and zumbrun

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 periodic traveling wave solutions of systems of viscous conservation laws in the generic case abstract extending previous results of oh zumbrun and johnson zumbrun we show that spectral stability implies linearized and nonlinear stability of spatially periodic traveling wave solutions of viscous systems of conservation laws for systems of generic type removing a restrictive assumption that wave speed be constant to first order along the manifold of nearby periodic solutions key to our analysis is a nonlinear cancellation estimate observed by johnson and zumbrun along with a detailed understanding of the whitham averaged system the latter motivates a careful analysis of the bloch perturbation expansion near zero frequency and suggests factoring out an appropriate translational modulation of the underlying wave allowing us to derive the sharpened low frequency estimates needed to close the nonlinear iteration arguments

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

stability of detonation profiles in the znd limit confirming a conjecture of lyng raoofi texier zumbrun we show that stability of strong detonation waves in the znd or small viscosity limit is equivalent to stability of the limiting znd detonation together with stability of the viscous profile associated with the component neumann shock moreover on bounded frequencies the nonstable eigenvalues of the viscous detonation wave converge to those of the limiting znd detonation while on frequencies of order one over viscosity they converge to one over viscosity times those of the associated viscous neumann shock this yields immediately a number of examples of instability and hopf bifurcation of reacting navier stokes detonations through the extensive numerical studies of znd stability in the detonation literature

metastability of solitary roll wave solutions of the st venant equations with viscosity we study by a combination of numerical and analytical evans function techniques the stability of solitary wave solutions of the st venant equations for viscous shallowwater ow down an incline and related models our main result is to exhibit examples of metastable solitary waves for the st venant equations with stable point spectrum indicating coherence of the wave prole but unstable essential spectrum indicating oscillatory convective instabilities shed in its wake we propose a mechanism based on dynamic spectrum of the wave prole by which a wave train of solitary pulses can stabilize each other by de amplication of convective instabilities as they pass through successive waves we present numerical time evolution studies supporting these conclusions which bear also on the possibility of stable periodic solutions close to the homoclinic for the closely related viscous jin xin model by contrast for which the essential spectrum is stable we show using the stability index of gardner zumbrun that solitary wave pulses are always exponentially unstable possessing point spectra with positive real part

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

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

a stability index for detonation waves in majda s model for reacting flow abstract using evans function techniques we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in siam j math anal 32 2001 929 commun pure appl math 51 7 1998 797 yielding useful necessary conditions for stability here we carry out the analysis in the context of the majda model a simplified model for reacting flow the method is extended to the full navier stokes equations of reacting flow in g lyng one dimensional stability of detonation waves doctoral thesis indiana university 2002 g lyng k zumbrun stability of detonation waves preprint 2003 the resulting stability condition is satisfied for all nondegenerate i e spatially exponentially decaying weak and strong detonations of the majda model in agreement with numerical experiments of siam j sci statist comput 7u 1986 1059 and analytical results of commun math phys 204 3 1999 551 commun math phys 202 3 1999 547 for a related model of majda and rosales we discuss also the role in the znd limit of degenerate subalgebraically decaying weak detonation and for a modified bump type ignition function deflagration profiles as discussed in siam j math anal 24 1993 968 siam j appl math 55 1995 175 for the full equations

application of differential quadrature to transport processes the methodology and numerical solution of problems concerning transport processes via the method of differential quadrature are presented application of the method is demonstrated by solving a simple one dimensional time dependent transient diffusion process involving an irreversible reaction without any flux across the end boundary in addition the same technique is used for the first time to the authors knowledge to solve a steady state problem for this purpose a convection diffusion problem involving an irreversible reaction is considered the demonstration is carried out in two ways 1 using the bellman et al technique which employs approximation formulas for higher order partial derivatives derived by iterating the linear quadrature approximation for the first order partial derivative and 2 using individual quadratures to approximate the partial derivatives of first as well as higher orders as suggested by mingle both approaches give the same results however the latter saves an appreciable amount of iterative computing effort despite the fact that it requires separate weighting coefficients for each individual quadrature since the technique of differential quadrature can produce solutions with sufficient accuracy even when using as few as three discrete points both the programming task and computational effort are alleviated considerably for these reasons the differential quadrature approach appears to be very practical in solving a variety of problems related to transport phenomena

spectral stability of noncharacteristic isentropic navier stokes boundary layers building on the work of barker humpherys lafitte rudd and zumbrun in the shock wave case we study stability of compressive or shock like boundary layers of the isentropic compressible navier stokes equations with law pressure by a combination of asymptotic ode estimates and numerical evans function computations our analytical results include convergence of the evans function in the shock and large amplitude limits and stability in the large amplitude limit the first rigorous stability result for other than the nearly constant case for all gamma geqq 1 together with these analytical results our numerical investigations indicate stability for 1 3 for all compressive boundary layers independent of amplitude save for inflow layers in the characteristic limit not treated expansive inflow boundary layers have been shown to be stable for all amplitudes by matsumura and nishihara using energy estimates besides the parameter of amplitude appearing in the shock case the boundary layer case features an additional parameter measuring displacement of the background profile which greatly complicates the resulting case structure inflow boundary layers turn out to have quite delicate stability in both large displacement shock and large amplitude limits necessitating the additional use of a mod two stability index studied earlier by serre and zumbrun in order to decide stability

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

l2 topological invariants of 3 manifolds we give results on thel2 betti numbers and novikov shubin invariants of compact manifolds especially 3 manifolds we first study the betti numbers and novikov shubin invariants of a chain complex of hilbert modules over a finite von neumann algebra we establish inequalities among the novikov shubin invariants of the terms in a short exact sequence of chain complexes our algebraic results along with some analytic results on geometric 3 manifolds are used to compute thel2 betti numbers of compact 3 manifolds which satisfy a weak form of the geometrization conjecture and to compute or estimate their novikov shubin invariants

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

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

stability of noncharacteristic boundary layers in the standing shock limit we investigate one and multi dimensional stability of noncharacteristic boundary layers in the limit approaching a standing planar shock wave u x1 x 1 0 obtaining necessary conditions of i weak stability of the limiting shock ii weak stability of the constant layer u u lim z z and iii nonnegativity of a modified lopatinski determinant similar to that of the inviscid shock case for lax 1 shocks we obtain equally simple sufficient conditions for p shocks p 1 the situation appears to be more complicated using these results we determine the stability of certain gas dynamical boundary layers generalizing earlier work of serre zumbrun and costanzino humphreys nguyen zumbrun

stability of multidimensional viscous shocks for symmetric systems with variable multiplicities we establish long time stability of multi dimensional viscous shocks of a general class of symmetric hyperbolic parabolic systems with variable multiplicities notably including the equations of compressible magnetohydrodynamics mhd in dimensions d 2 this extends the existing result established by k zumbrun for systems with characteristics of constant multiplicity to the ones with variable multiplicity yielding the first such a stability result for fast mhd shocks at the same time we are able to drop a technical assumption on structure of the so called glancing set that was necessarily used in previous analyses the key idea to the improvements is to introduce a new simple argument for obtaining a l l resolvent bound in low frequency regimes by employing the recent construction of degenerate kreiss symmetrizers by o gues g metivier m williams and k zumbrun thus at the low frequency resolvent bound level our analysis gives an alternative to the earlier pointwise green s function approach of k zumbrun high frequency solution operator bounds have been previously established entirely by nonlinear energy estimates

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

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

boundary layer stability in real vanishing viscosity limit in the previous paper 20 an evans function machinery for the study of boundary layer stability was developed there the analysis was restricted to strongly parabolic perturbations that is to an approximation of the form ut f u x b u ux x 1 with an elliptic matrix b however real models like the navier stokes approximation of the euler equations for a gas flow involve incompletely parabolic perturbations b is not invertible in general

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

global well posedness of the cauchy problem for the navier stokes equations of nonisentropic flow with discontinuous initial data abstract we prove the global existence uniqueness and continuous dependence on initial data for discontinuous solutions of the navier stokes equations for nonisentropic compressible flow in one space dimension a great deal of information is obtained concerning the qualitative behavior of the solution including an analysis of the convection and evolution of jump discontinuities the derivation of sharp rates of smoothing and the l asymptotic behavior

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

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

pointwise asymptotic behavior of modulated periodic reaction diffusion waves abstract by working with the periodic resolvent kernel and the bloch decomposition we establish pointwise bounds for the green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction diffusion equations with our linearized estimates together with a nonlinear iteration scheme developed by johnson zumbrun we obtain l p behavior p 1 of a nonlinear solution to a perturbation equation of a reaction diffusion equation with respect to initial data in l 1 h 2 recovering and slightly sharpening results obtained by schneider using weighted energy and renormalization techniques we obtain also pointwise nonlinear estimates with respect to two different initial perturbations u 0 e 0 e x 2 m u 0 h 2 e 0 and u 0 e 0 1 x r r 2 u 0 h 2 e 0 respectively e 0 0 sufficiently small and m 1 sufficiently large showing that behavior is that of a heat kernel these pointwise bounds have not been obtained elsewhere and do not appear to be accessible by previous techniques

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