uniform positivity improving property sobolev inequalities and spectral gaps abstract we introduce a certain kind of strong ergodicity condition to study the existence of spectral gap for markov generator we can estimate the spectral gap using the ergodicity condition and a sobolev type inequality we apply our results to the dirichlet form on wiener spaces riemannian manifolds and loop spaces

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

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

behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations we establish nonlinear stability and asymptotic behavior of traveling periodic waves of viscous conservation laws under localized perturbations or nonlocalized perturbations asymptotic to constant shifts in phase showing that long time behavior is governed by an associated second order formal whitham modulation system a key point is to identify the way in which initial perturbations translate to initial data for this formal system a task accomplished by detailed estimates on the linearized solution operator about the background wave notably our approach gives both a common theoretical treatment and a complete classification in terms of phase coupling or decoupling of general systems of conservation or balance laws encompassing cases that had previously been studied separately or not at all at the same time our refined description of solutions gives the new result of nonlinear asymptotic stability with respect to localized perturbations in the phase decoupled case further distinguishing behavior in the different cases an interesting technical aspect of our analysis is that for systems of conservation laws the whitham modulation description is of system rather than scalar form as a consequence of which renormalization methods such as have been used to treat the reaction diffusion case in general do not seem to apply

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

bounds on the spectrum for markov chains and markov processes a generalization of cheeger s inequality we prove a general version of cheeger s inequality for discrete time markov chains and continuous time markovian jump processes both reversible and nonreversible with general state space we also prove a version of cheeger s inequality for markov chains and processes with killing as an application we prove l 2 exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs

multi way dual cheeger constants and spectral bounds of graphs we introduce a set of multi way dual cheeger constants and prove universal higher order dual cheeger inequalities for eigenvalues of normalized laplace operators on weighted finite graphs our proof proposes a new spectral clustering phenomenon deduced from metrics on real projective spaces we further extend those results to a general reversible markov operator and find applications in characterizing its essential spectrum

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

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

existence and stability of viscoelastic shock profiles we investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by antman and malek madani we establish that the resulting equations fall into the class of symmetrizable hyperbolic parabolic systems hence spectral stability implies linearized and nonlinear stability with sharp rates of decay the new contributions are treatment of the compressible case formulation of a rigorous nonlinear stability theory including verification of stability of small amplitude lax shocks and the systematic incorporation in our investigations of numerical evans function computations determining stability of large amplitude and nonclassical type shock profiles

order conditions for numerical methods for partitioned ordinary differential equations motivated by the consideration of runge kutta formulas for partitioned systems the theory of p series is studied this theory yields the general structure of the order conditions for numerical methods for partitioned systems and in addition for nystrom methods fory f y y for rosenbrock type methods with inexact jacobian w methods it is a direct generalization of the theory of butcher series 7 8 in a later publication the theory ofp series will be used for the derivation of order conditions for runge kutta type methods for volterra integral equations 1

implicit runge kutta methods of optimal order for volterra integro differential equations implicit runge kutta methods with m stages and optimal order p 2m for the approximate solution of volterra integro differential equations can be viewed as fully dis cretized collocation methods in certain polynomial spline spaces the choice of the quadrature formulas needed for the full discretization of the collocations is investigated and it is shown that in contrast to ordinary differential equations there exist for fixed m several optimal methods 1 introduction in this paper we shall be concerned with the approximate solution of the initial value problem for a nonlinear first order volterra integro differential equation

runge kutta theory for volterra integrodifferential equations the present paper develops the theory of general runge kutta methods for volterra integrodifferential equations the local order is characterized in terms of the coefficients of the method we investigate the global convergence of mixed and extended runge kutta methods and give results on asymptotic error expansions in a further section we construct examples of methods up to order 4

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

analysis on wiener spaces ii differential forms abstract the author gives the notions of p forms and p forms and shows some properties of them in particular a vanishing theorem of de rham cohomology of poincare type also he shows the hypoellipticity of certain elliptic operators on wiener space and a theorem of poinare inequality type

stability and instability of solitary waves of the fifth order kdv equation a numerical framework abstract the spectral problem associated with the linearization about solitary waves of the generalized fifth order kdv equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to eigenvalues a numerical framework based on a fast robust shooting algorithm on exterior algebra spaces is introduced the complete algorithm has several new features including a rigorous numerical algorithm for choosing starting values a new method for numerical analytic continuation of starting vectors the role of the grassmannian g 2 c 5 in choosing the numerical integrator and the role of the hodge star operator for relating 2 c 5 and 3 c 5 and deducing a range of numerically computable forms for the evans function the algorithm is illustrated by computing the stability and instability of solitary waves of the fifth order kdv equation with polynomial nonlinearity

convex entropy hopf bifurcation and viscous and inviscid shock stability we consider by a combination of analytical and numerical techniques some basic questions regarding the relations between inviscid and viscous stability and the existence of a convex entropy specifically for a system possessing a convex entropy in particular for the equations of gas dynamics with a convex equation of state we ask 1 can inviscid instability occur 2 can viscous instability not detected by inviscid theory occur 3 can there occur the necessarily viscous effect of hopf bifurcation or galloping instability and perhaps most important from a practical point of view 4 as shock amplitude is increased from the stable weak amplitude limit can there occur a first transition from viscous stability to instability that is not detected by inviscid theory we show that 1 does occur for strictly hyperbolic genuinely nonlinear gas dynamics with certain convex equations of state while 2 and 3 do occur for an artifically constructed system with convex viscosity compatible entropy we do not know of an example for which 4 occurs leaving this as a key open question in viscous shock theory related to the principal eigenvalue property of sturm liouville and related operators in analogy with and partly proceeding close to the analysis of smith on non uniqueness of the riemann problem we obtain convenient criteria for shock in stability in the form of necessary and sufficient conditions on the equation of state

uniformly integrable operators and large deviations for markov processes abstract in this paper we introduce and investigate the notion of uniformly integrable operators on l p e its relations to classical compactness and hypercontractivity are exhibited several consequences of this notion are established such as perron frobenius type theorems independence on p of the spectral radius in l p p continuity of spectral radius and especially the existence of spectral gap in the irreducible case we also present some infinitesimal criteria ensuring the uniform integrability of a positive semigroup or of its resolvent for a essentially irreducible markov process we show that the uniform integrability in l p of some type of resolvent associated with the transition semigroup implies the large deviation principle of level 3 with some rate function given by a modified donsker varadhan entropy functional we also prove that the uniform integrability condition becomes even necessary in the symmetric case finally we present several applications of our results to feynman kac semigroups to the thermodynamical limits of grand ensembles and to non symmetric markov processes given by girsanov s formula

stability of travelling waves for non convex scalar viscous conservation laws travelling waves of a viscous conservation law so called viscous profiles are shown to be stable in polynomialy weighted l spaces there is no assumption of convexity on the nonlinear term and thus earlier results of il in and ole nik sattinger and kawashima and matsumura are generalized the method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates are derived for the semigroup through a new technique for estimating the resolvent 1993 john wiley sons inc

poincare type inequalities for singular stable like dirichlet forms abstract this paper is concerned with a class of singular stable like dirichlet forms on r d which are generated by d independent copies of a one dimensional symmetric stable process and whose levy jump kernel measure is concentrated on the union of the coordinate axes explicit and sharp criteria for poincare inequality super poincare inequality and weak poincare inequality of such singular dirichlet forms are presented when the reference measure is a product measure on r d we also consider the entropy inequality for the associated dirichlet forms which is similar to the log sobolev inequality for local dirichlet forms and enjoys the tensorisation property

runge kutta theory for volterra integral equations of the second kind the present paper develops the theory of general runge kutta methods for volterra integral equations of the second kind the order conditions are derived by using the theory of p series which for our problem reduces to the theory of v series these results are then applied to two special classes of runge kutta methods introduced by pouzet and by bel tyukov

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

natural continuous extensions of runge kutta methods the present paper develops a theory of natural continuous extensions nces for the discrete approximate solution of an ode given by a runge kutta process these nces are defined in such a way that the continuous solutions furnished by the one step collocation methods are included

spectral gaps for hyperbounded operators abstract we consider a positive and power bounded linear operator t on l p over a finite measure space and prove that if t l p l q for some q p then the essential spectral radius of t is strictly smaller than 1 as a special case we obtain a recent result of miclo who proved this assertion for self adjoint ergodic markov operators in the case p 2 and thereby solved a long open problem of simon and hoegh krohn our methods draw a connection between spectral theory and the geometry of banach spaces they rely on a result going back to groh that encodes spectral gap properties via ultrapowers and on the fact that an infinite dimensional l p space cannot be isomorphic to an l q space for q p we also prove a number of variations of our main result i it follows from theorems of lotz and martinez that the condition t l p l q can be replaced with the weaker assumption that t maps the positive part of the l p unit ball into a uniformly p integrable set ii while it is known that the positivity assumption on t cannot in general be omitted we show that we can replace it with the assumption that t is contractive both on l p and on l q iii we prove a version of the theorem which allows us under appropriate circumstances to also consider non finite measures spaces iv our result also has a uniform version there exists an upper bound c 0 1 for the essential spectral radius of t where c depends on certain quantitative properties of t l p and l q

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

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

natural continuous extensions of runge kutta methods for volterra integral equations of the second kind and their applications we consider a very general class of runge kutta methods for the numerical solution of volterra integral equations of the second kind which includes as special cases all the more important methods which have been considered in the literature the main purpose of this paper is to define and prove the existence of the natural continuous extensions nce s of runge kutta methods i e piecewise polynomial functions which extend the approximation at the grid points to the whole interval of integration the particular properties required of the nce s allow us to construct the tail approximations which are quite efficient in terms of kernel evaluations

numerical testing of the stability of viscous shock waves a new theoretical evans function condition is used as the basis of a numerical test of viscous shock wave stability accuracy of the method is demonstrated through comparison against exact solutions a convergence study and evaluation of approximate error equations robustness is demonstrated by applying the method to waves for which no current analytic results apply highly nonlinear waves from the cubic model and strong shocks from gas dynamics an interesting aspect of the analysis is the need to incorporate features from the analytic evans function theory for purposes of numerical stability for example we find it necessary for numerical accuracy to solve odes on the space of wedge products

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

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

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

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

connectivity of phase boundaries in strictly convex domains we consider equilibria arising in a model for phase transitions which correspond to stable critical points of the constrained variational problem inf int omega u dx m int omega frac 1 varepsilon w u frac varepsilon 2 big nabla u big 2 dx here w is a double well potential and omega subset r n is a strictly convex domain for e small this is closely related to the problem of partitioning into two subdomains of fixed volume where the subdomain boundaries correspond to the transitional boundary between phases motivated by this geometry problem we show that in a strictly convex domain stable critical points of the original variational problem have a connected thin transition layer separating the two phases this relates to work in gm where special geometries such as cylindrical domains were treated and is analogous to the results in cho which show that in a convex domain stable critical points of the corresponding unconstrained problem are constant the proof of connectivity employs tools from geometric measure theory including the co area formula and the isoperimetric inequality on manifolds the thinness of the transition layer follows from a separate calculation establishing spatial decay of solutions to the pure phases

stability analysis of the de hoog and weiss implicit runge kutta methods for the volterra integral and integrodifferential equations abstract the stability of the de hoog and weiss runge kutta methods is analyzed for the volterra integrodifferential equations with respect to the basic test equation and for volterra integral equations with respect to the linear convolution equation stability regions are determined for some choices of the parameters

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

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

plates with incompatible prestrain we study effective elastic behavior of the incompatibly prestrained thin plates where the prestrain is independent of thickness and uniform through the plate s thickness h we model such plates as three dimensional elastic bodies with a prescribed pointwise stress free state characterized by a riemannian metric g and seek the limiting behavior as h to 0 we first establish that when the energy per volume scales as the second power of h the resulting gamma limit is a kirchhoff type bending theory we then show the somewhat surprising result that there exist non immersible metrics g for whom the infimum energy per volume scales smaller than h2 this implies that the minimizing sequence of deformations carries nontrivial residual three dimensional energy but it has zero bending energy as seen from the limit kirchhoff theory perspective another implication is that other asymptotic scenarios are valid in appropriate smaller scaling regimes of energy we characterize the metrics g with the above property showing that the zero bending energy in the kirchhoff limit occurs if and only if the riemann curvatures r1213 r1223 and r1212 of g vanish identically we illustrate our findings with examples of particular interest is an example where g 2 times 2 the two dimensional restriction of g is flat but the plate still exhibits the energy scaling of the foppl von karman type finally we apply these results to a model of nematic glass including a characterization of the condition when the metric is immersible for g id 3 gamma n otimes n given in terms of the inhomogeneous unit director field distribution n in mathbb r 3

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

a lax shock profile satisfying a sufficient condition of spectral instability abstract the aim of this paper is to construct a spectrally unstable lax shock profile of the semi linear approximation of a 2 2 system of conservation laws and to show numerically the instability thanks to an appropriate scheme

natural continuous extensions of runge kutta methods for volterra integrodifferential equations in this paper we deal with a very general class of runge kutta methods for the numerical solution of volterra integrodifferential equations our main contribution is the development of the theory of natural continuous extensions nces i e piecewice polynomial functions which interpolate the values given by the rk method at the mesh points the particular features of these nces allow us to construct tail approximations which are quite efficient since they require a minimal number of kernel evaluations

criteria of spectral gap for markov operators let e f be a probability space and let p be a markov operator on l2 with 1 a simple eigenvalue such that p i e is an invariant probability measure of p then p 12 p p has a spectral gap i e 1 is isolated in the spectrum of p if and only if p limr sup f2 1 f pf r 1 this strengthens a conjecture of simon and hoegh krohn on the spectral gap for hyperbounded operators solved recently by l miclo in 10 consequently for a symmetric conservative irreducible dirichlet form on l2 a poincare log sobolev type inequality holds if and only if so does the corresponding defective inequality extensions to sub markov operators and non conservative dirichlet forms are also presented

weak poincare inequalities and l2 convergence rates of markov semigroups abstract in order to describe l 2 convergence rates slower than exponential the weak poincare inequality is introduced it is shown that the convergence rate of a markov semigroup and the corresponding weak poincare inequality can be determined by each other conditions for the weak poincare inequality to hold are presented which are easy to check and which hold in many applications the weak poincare inequality is also studied by using isoperimetric inequalities for diffusion and jump processes some typical examples are given to illustrate the general results in particular our results are applied to the stochastic quantization of field theory in finite volume moreover a sharp criterion of weak poincare inequalities is presented for poisson measures on configuration spaces

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

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

functional inequalities and spectrum estimates the infinite measure case abstract as a continuation of 13 where a poincare type inequality was introduced to study the essential spectrum on the l 2 space of a probability measure this paper provides a modification of this inequality so that the infimum of the essential spectrum is well described even if the reference measure is infinite high order eigenvalues as well as the corresponding semigroup are estimated by using this new inequality criteria of the inequality and estimates of the inequality constants are presented finally some concrete examples are considered to illustrate the main results in particular estimates of high order eigenvalues obtained in this paper are sharp as checked by two examples on the euclidean space

normal modes and nonlinear stability behaviour of dynamic phase boundaries in elastic materials this paper considers an ideal nonthermal elastic medium described by a stored energy function w it studies time dependent configurations with subsonically moving phase boundaries across which in addition to the jump relations of rankine hugoniot type expressing conservation some kinetic rule g acts as a two sided boundary condition the paper establishes a concise version of a normal modes determinant that characterizes the local in time linear and nonlinear in stability of such patterns specific attention is given to the case where w has two local minimizers ua ub which can coexist via a static planar phase boundary being dynamic perturbations of such interesting configurations this paper shows that the stability behaviour of corresponding almost static phase boundaries is uniformly controlled by an explicit expression that can be determined from derivatives of w and g at ua and ub

an efficient shooting algorithm for evans function calculations in large systems abstract in evans function computations of the spectra of asymptotically constant coefficient linear operators a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ode for small systems a fast shooting algorithm may be obtained by representing subspaces as single exterior products j c alexander r sachs linear instability of solitary waves of a boussinesq type equation a computer assisted computation nonlinear world 2 4 1995 471 507 l q brin numerical testing of the stability of viscous shock waves ph d thesis indiana university bloomington 1998 l q brin numerical testing of the stability of viscous shock waves math comp 70 235 2001 1071 1088 l q brin k zumbrun analytically varying eigenvectors and the stability of viscous shock waves in seventh workshop on partial differential equations part i 2001 rio de janeiro mat contemp 22 2002 19 32 t j bridges g derks g gottwald stability and instability of solitary waves of the fifth order kdv equation a numerical framework physica d 172 1 4 2002 190 216 for large systems however the dimension of the exterior product space quickly becomes prohibitive growing as n k where n is the dimension of the system written as a first order ode and k typically n 2 is the dimension of the subspace we resolve this difficulty by the introduction of a simple polar coordinate algorithm representing pure monomial products as scalar multiples of orthonormal bases for which the angular equation is a numerically optimized version of the continuous orthogonalization method of drury davey a davey an automatic orthonormalization method for solving stiff boundary value problems j comput phys 51 2 1983 343 356 l o drury numerical solution of orr sommerfeld type equations j comput phys 37 1 1980 133 139 and the radial equation is evaluable by quadrature notably the polar coordinate method preserves the important property of analyticity with respect to parameters

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

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

stability properties of interpolants for runge kutta methods in solving initial value problems for ordinary differential equations by means of a discrete method we often need to know the solution on a set of points that differs from the grid systems with driving equation dense output discontinuity problems delay equations etc the nonnodal approximations are generally obtained by local interpolation and different features can be requested from the interpolants according to the problem treated in this paper we introduce the concept of stable interpolant for runge kutta methods roughly speaking a stable interpolant maintains the stability properties of the runge kutta method itself the lack of such stability makes the continuous extension not reliable especially in solving stiff equations

estimates for invariant probability measures of degenerate spdes with singular and path dependent drifts in terms of a nice reference probability measure integrability conditions on the path dependent drift are presented for infinite dimensional degenerate pdes to have regular positive solutions to this end the corresponding stochastic partial differential equations are proved to possess the weak existence and uniqueness of solutions as well as the existence uniqueness and entropy estimates of invariant probability measures when the reference measure satisfies the log sobolev inequality sobolev estimates are derived for the density of invariant probability measures some results are new even for non degenerate sdes with path independent drifts the main results are applied to nonlinear functional spdes and degenerate functional sdes spdes

on hyperboundedness and spectrum of markov operators consider an ergodic markov operator m reversible with respect to a probability measure mu on a general measurable space it is shown that if m is bounded from mathbb l 2 mu to mathbb l p mu where p 2 then it admits a spectral gap this result answers positively a conjecture raised by hoegh krohn and simon j funct anal 9 121 80 1972 in the more restricted semi group context the proof is based on isoperimetric considerations and especially on cheeger inequalities of higher order for weighted finite graphs recently obtained by lee et al proceedings of the 2012 acm symposium on theory of computing 1131 1140 acm new york 2012 it provides a quantitative link between hyperboundedness and an eigenvalue different from the spectral gap in general in addition the usual cheeger inequality is extended to the higher eigenvalues in the compact riemannian setting and the exponential behaviors of the small eigenvalues of witten laplacians at small temperature are recovered

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

absolute stability of explicit runge kutta nystrom methods for y ƒ x y y abstract we examine absolute stability of s stage explicit runge kutta nystrom r k n methods of order s for s 2 3 4 for y x y y by applying these methods to the test equation y 2 y 2y 0 0 we show the existence of r k n methods of orders two three and four possessing intervals of absolute stability as large as that of explicit runge kutta r k methods of respective orders

oscillatory instability of traveling waves for a kdv burgers equation the stability of traveling wave solutions of a generalization of the kdv burgers equation o u u p o u 03u a o u is studied as the parameters p and a are varied the eigenvalue problem for the linearized evolution of perturbations is analyzed by numerically computing evans function d a an analytic function whose zeros correspond to discrete eigenvalues in particular the number of unstable eigenvalues in the complex plane is evaluated by computing the winding number of d a analytical and numerical evidence suggests that a hopf bifurcation occurs for oscillatory traveling wave profiles in certain parameter ranges dynamic simulations suggest that the bifurcation is subcritical no stable time periodic solution is found

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

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