landis conjecture for general second order elliptic equations with singular lower order terms in the plane abstract in this article we study the order of vanishing and a quantitative form of landis conjecture in the plane for solutions to second order elliptic equations with variable coefficients and singular lower order terms precisely we let a be real valued bounded and elliptic but not necessary symmetric or continuous and we assume that v and w i are real valued and belong to l p and l q i respectively we prove that if u is a real valued bounded and normalized solution to an equation of the form div a u w 1 u w 2 u v u 0 in b d then under suitable conditions on the lower order terms for any r sufficiently small the following order of vanishing estimate holds u l b r r c m where m depends on the lebesgue norms of the lower order terms in a number of settings a scaling argument gives rise to a quantitative form of landis conjecture inf z 0 r u l b 1 z 0 exp c r log r where depends on p q 1 and q 2 the integrability assumptions that we impose on v and w i are nearly optimal in view of a scaling argument we use the theory of elliptic boundary value problems to establish the existence of positive multipliers associated to the elliptic equation then the proofs rely on transforming the equations to beltrami systems and applying a generalization of hadamard s three circle theorem

stability of viscous shock waves associated with a system of nonstrictly hyperbolic conservations laws we introduce a simple model of two conservation laws which is strictly hyperbolic except for a degenerate parabolic line in the state space besides classical shock waves it also exhibits overcompressive marginal overcompressive and marginal undercompressive shock waves our purpose is to study the behavior of the corresponding viscous waves in particular the manner in which these waves are stable there are several basic differences between classical shock waves and other types of shock waves a perturbation of an overcompressive shock wave gives rise to a new wave monotone marginal overcompressive waves behave distinctly from the nonmonotone ones analytical techniques used in our study include characteristic energy and weighted energy methods and nonlinear superposition through time invariants although we carry out our analysis for a simple model the general phenomena would hold for overcompressive waves which occur in other physical models

l2 contraction of large planar shock waves for multi dimensional scalar viscous conservation laws abstract we consider a l 2 contraction a l 2 type stability of large viscous shock waves for the multi dimensional scalar viscous conservation laws up to a suitable shift by using the relative entropy methods quite different from the previous results we find a new way to determine the shift function which depends both on the time and space variables and solves a viscous hamilton jacobi type equation with source terms moreover we do not impose any conditions on the anti derivative variables of the perturbation around the shock profile more precisely it is proved that if the initial perturbation around the viscous shock wave is suitably small in l 2 norm then the l 2 contraction holds true for the viscous shock wave up to a suitable shift function note that bv norm or the l norm of the initial perturbation and the shock wave strength can be arbitrarily large furthermore as the time t tends to infinity the l 2 contraction holds true up to a spatially homogeneous time dependent shift function in particular if we choose some special initial perturbations then l 2 convergence of the solutions towards the associated shock profile can be proved up to a time dependent shift

weak and strong detonation profiles for a qualitative model abstract in this article the unidirectional travelling wave fronts for a qualitative model in combustion are studied the model is considered for a simple r p kind of chemical reaction by proving a general theorem of ode s the weak and strong detonations for exothermic reactions are shown to exist the uniqueness of these waves are also considered

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

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

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

on a parabolic hyperbolic chemotaxis system with discontinuous data well posedness stability and regularity abstract the global dynamics and regularity of parabolic hyperbolic systems is an interesting topic in pdes due to the coupling of competing dissipation and hyperbolic effects this paper is concerned with the cauchy problem of a parabolic hyperbolic system derived from a chemotaxis model describing the dynamics of the initiation of tumor angiogenesis it is shown that as time tends to infinity the cauchy problem with large amplitude discontinuous data admit global weak solutions which converge to a constant state resp a viscous shock wave if the asymptotic states of initial values at far field are equal resp unequal our results improve the previous results where initial value was required to be continuous and have small amplitude numerical simulations are performed to verify our analytical results illustrate the possible regularity of solutions and speculate the minimal regularity of initial data required to obtain the smooth classical solutions of the concerned parabolic hyperbolic system

stability of viscous scalar shock fronts in several dimensions we prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions the proof uses the integrated equation and an effective equation for the motion of the front itself we derive energy estimates that balance terms from the integrated equation with terms from the front motion equation in this paper we prove that viscous shock profiles for scalar conservation laws are stable in two or more space dimensions these multidimensional stability questions are separate from their one dimensional analogues because of the possibility of transverse instabilities such as those that occur in combustion fronts lu and in shock waves with phase changes the proof here is a rigorous version of arguments that are used to derive effective equations such as the kuramoto sivashinsky equation to describe the behavior of fronts the one dimensional stability for scalar conservation laws was proven by ii in and oleinik io using the integrated equation see below and a max imum principle another proof based on weighted norms and spectral theory for the linearized problem was given by sattinger s the multidimensional stability proof below has more in common with the stability proofs for systems of conservation laws in one space dimension begun by kawashima and mat sumura km and goodman go and completed by liu li these proofs use l2 energy estimates for the integrated equation we consider equations of the form ut f u x g u y uxx uyy where f u is a strictly convex function of u

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

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

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

on the linearized stability of viscous shock profiles for systems of conservation laws abstract this paper is concerned with the linearized stability of traveling wave solutions for systems of viscous hyperbolic conservation laws the main purpose is to show that for a given traveling wave with shock profile from any characteristic family there exists an appropriate weighted norm space such that the traveling wave is exponentially stable in this space as a consequence if the initial disturbance has average zero and decays exponentially fast as x then the corresponding solution of the linearized equation decays to zero exponentially fast in t on any compact interval in x the proof is given by applying an elementary weighted characteristic energy method to the integrated linearized system based on the underlying wave structure

on the fractional landis conjecture abstract in this paper we study a landis type conjecture for fractional schrodinger equations of fractional power s 0 1 with potentials we discuss both the cases of differentiable and non differentiable potentials on the one hand it turns out for differentiable potentials with some a priori bounds if a solution decays at a rate e x 1 then this solution is trivial on the other hand for s 1 4 1 and merely bounded non differentiable potentials if a solution decays at a rate e x with 4 s 4 s 1 then this solution must again be trivial remark that when s 1 4 s 4 s 1 4 3 which is the optimal exponent for the standard laplacian for the case of non differentiable potentials and s 1 4 1 we also derive a quantitative estimate mimicking the classical result by bourgain and kenig

quantitative uniqueness estimates for the general second order elliptic equations abstract in this paper we study quantitative uniqueness estimates of solutions to general second order elliptic equations with magnetic and electric potentials we derive lower bounds of decay rate at infinity for any nontrivial solution under some general assumptions the lower bounds depend on asymptotic behaviors of magnetic and electric potentials the proof is carried out by the carleman method and bootstrapping arguments

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

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

asymptotic stability of relaxation shock profiles for hyperbolic conservation laws abstract this paper studies the asymptotic stability of traveling relaxation shock profiles for hyperbolic systems of conservation laws under a stability condition of subcharacteristic type the large time relaxation dynamics on the level of shocks is shown to be determined by the equilibrium conservation laws the proof is due to the energy principle using the weighted norms the interaction of waves from various modes is treated by imposing suitable weight matrix

global stability of viscous contact wave for 1 d compressible navier stokes equations the viscous contact waves for one dimensional compressible navier stokes equations has recently been shown to be asymptotically stable the stability results are called local stability or global stability depending on whether the norms of initial perturbations are small or not up to now local stability results toward viscous contact waves of compressible navier stokes equations have been well established see huang et al 2006 2008 2009 9 10 7 but there are few results for the global stability in the case of cauchy problem which is the purpose of this paper the proof is based on an elementary energy method using an inequality concerning the heat kernel see lemma 1 of huang et al 2010 7

on the stability of finite element methods for shock waves this paper we study the large time asymptotic stability of solutions for systems of nonlinear viscous conservation laws of the form 1 1 u t f u x u xx x 2 r i t 0 u 2 r i u delta 0 u 0 delta we treat systems which are strictly hyperbolic such systems possess a smooth travelling wave solution which is called a viscous p shock wave solution u x t oe x gamma oet x sigma1 oe x u sigma provided that the shock strength ffl j ju gamma u gamma j is small 19 the constant states u sigma and the wave speed oe are related by the rankine hugoniot condition 1 3a f u gamma gamma f u oe u gamma gamma u

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

regularity of radial minimizers and extremal solutions of semilinear elliptic equations we consider a special class of radial solutions of semilinear equations u g u in the unit ball of rn it is the class of semi stable solutions which includes local minimizers minimal solutions and extremal solutions we establish sharp pointwise lq and wk q estimates for semi stable radial solutions our regularity results do not depend on the specific nonlinearity g among other results we prove that every semi stable radial weak solution u h01 is bounded if n 9 for every g and belongs to h3 w3 2 in all dimensions n for every g increasing and convex the optimal regularity results are strongly related to an explicit exponent which is larger than the critical sobolev exponent

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

convergence to travelling fronts of solutions of the p system with viscosity in the presence of a boundary we study the asymptotic behavior as time goes to infinity of solutions to the initial boundary value problem on the half space r for a one dimensional model system for the isentropic flow of a compressible viscous gas the so called p system with viscosity as boundary conditions we prescribe the constant state at infinity and require that the velocity be zero at the boundary x 0 when the velocity at infinity is negative and satisfies a condition on the magnitude we prove that if the initial data are suitably close to those for the corresponding outgoing viscous shock profile which is suitably far from the boundary then a unique solution exists globally in time and tends toward the properly shifted viscous shock profile as the time goes to infinity the proof is given by an elementary energy method

existence of chapman jouguet detonation for a viscous combustion model in siam j appl math 41 1981 70 93 majda proposed a model for the interaction between chemical reactions and compressible fluid dynamics this model is a low mach number limit of the one component reactive navier stokes equations siam j appl math 43 1983 1086 1118 and was extended to the case where the diffusion coefficient is positive by larrouturou nonlinear anal 77 2001 405 418 in this paper the existence of a one dimensional chapman jouguet detonation wave or equivalently a heteroclinic orbit for the extended model is proven the proof is based on an application of topological arguments to a system of ordinary differential equations which is obtained from the partial differential equations describing the interaction

gaussian decay of harmonic oscillators and related models abstract we prove that the decay of the eigenfunctions of harmonic oscillators uniform electric or magnetic fields is not stable under 0 order complex perturbations even if bounded of these hamiltonians in the sense that we can produce solutions to the evolutionary schrodinger flows associated to the hamiltonians with a stronger gaussian decay at two distinct times we then characterize in a quantitative way the sharpest possible gaussian decay of solutions as a function of the oscillation frequency or the strength of the field depending on the hamiltonian which is considered this is connected to the hardy s uncertainty principle for free schrodinger evolutions

asymptotic stability of traveling wave solutions of systems for one dimensional gas motion the asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics 1 the solution of a scalar conservation law with viscosity approaches the traveling wave solution at the ratet for some 0 ast provided that the initial disturbance is small and of integral zero and in addition decays at an algebraic rate for x 2 the traveling wave solution with nishida and smoller s condition of the system of a viscous heat conductive ideal gas is asymptotically stable provided the initial disturbance is small and of integral zero 3 the traveling wave solution with weak shock profile of the broadwell model system of the boltzmann equation is asymptotically stable provided the initial disturbance is small and its hydrodynamical moments are of integral zero each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one the property of integral zero of the initial disturbance plays a crucial role in this procedure

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

stability of 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

stability of a composite wave of two viscous shock waves for the full compressible navier stokes equation in this paper we investigate the asymptotic stability of a composite wave consisting of two viscous shock waves for the full compressible navier stokes equation by introducing a new linear diffusion wave special to this case we successfully prove that if the strengths of the viscous shock waves are suitably small with same order and also the initial perturbations which are not necessarily of zero integral are suitably small the unique global solution in time to the full compressible navier stokes equation exists and asymptotically tends toward the corresponding composite wave whose shifts in space of two viscous shock waves are uniquely determined by the initial perturbations we then apply the idea to study a half space problem for the full compressible navier stokes equation and obtain a similar result

unique continuation for elliptic operators a geometric variational approach dans un sous ensemble connexe ouvert de r n n 3 on considere l equation d ordre 2 lu div a x u b x u v x u 0 ou a x a id j x est une matrice symetrique reelle on etudie les proprietes d unicite de l operateur

stability of rarefaction waves in viscous media we study the time asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one dimensional viscous media with strictly hyperbolic flux functions our main result is to show that solutions of perturbed rarefaction data converge to an approximate burgers rarefaction wave for initial perturbations w 0 with small mass and localized as w 0 x mathcal o x 1 the proof proceeds by iteration of a pointwise ansatz for the error using integral representations of its various components based on green s functions we estimate the green s functions by careful use of the hopf cole transformation combined with a refined parametrix method as a consequence of our method we also obtain rates of decay and detailed pointwise estimates for the error this pointwise method has been used successfully in studying stability of shock and constant state solutions new features in the rarefaction case are time varying coefficients in the linearized equations and error waves of unbounded mass mathcal o log t these diffusion waves have amplitude mathcal o t 1 2logt in linear degenerate transversal fields and mathcal o t 1 2logt in genuinely nonlinear transversal fields a distinction which is critical in the stability proof

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

quantitative unique continuation for schrodinger operators abstract we investigate the quantitative unique continuation properties of solutions to second order elliptic equations with singular lower order terms the main theorem presents a quantification of the strong unique continuation property for v that is for any non trivial u that solves u v u 0 in some open connected subset of r n we estimate the vanishing order of solutions in terms of the l t norm of v our results apply to all t n 2 and n 3 with these maximal order of vanishing estimates we employ a scaling argument to produce quantitative unique continuation at infinity estimates for global solutions to u v u 0 to handle v l t for every t n 2 we prove a novel l p l q carleman estimate by interpolating a known l p l 2 estimate with a new endpoint carleman estimate this new carleman estimate may also be used to establish improved order of vanishing estimates for equations with a first order term those of the form u w u v u 0

adaptive finite element methods for conservation laws based on a posteriori error estimates we prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension and design corresponding adaptive methods the proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem together with the galerkin orthogonality of the finite element method the strong stability estimate uses the entropy condition for the system in an essential way 1995 john wiley sons inc

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 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

nonlinear stability of rarefaction waves for the compressible navier stokes equations with zero heat conductivity abstract this paper is concerned with the time asymptotic nonlinear stability of rarefaction waves to the cauchy problem of the one dimensional compressible navier stokes equations with zero heat conductivity under the assumption that the unique global entropy solution to the resulting riemann problem of the corresponding compressible euler equations consists of rarefaction waves only then if both the initial perturbation and the strengths of rarefaction waves are assumed to be suitably small we show that its cauchy problem admits a unique global solution which tends time asymptotically toward the rarefaction waves this result is proved by using the elementary energy method and the argument developed by kawashima and okada 1982

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m

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

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

sharp estimates for semi stable radial solutions of semilinear elliptic equations this paper is devoted to the study of semi stable radial solutions u h1 b1 of u g u in b1 0 where g c1 r is a general nonlinearity and b1 is the unit ball of rn we establish sharp pointwise estimates for such solutions as an application of these results we obtain optimal pointwise estimates for the extremal solution and its derivatives up to order three of the semilinear elliptic equation u f u posed in b1 with dirichlet data u b1 0 where f is a continuous positive nondecreasing and convex function on 0 such that f s s as s in addition we provide for n 10 a large family of semi stable radially decreasing unbounded h1 b1 solutions

large time behavior of solutions of lax friedrichs finite difference equations for hyperbolic systems of conservation laws the author studies the large time behavior of discrete solutions of the lax friedrichs finite difference equations for hyperbolic systems of conservation laws the initial data considered here are small and tend to a constant state at x plus minus infinity he shows that the solutions tend to the discrete diffusion waves at the rate o t sup minus 3 4 1 2p sigma in l sup 0 1 le p le infinity with sigma 0 being an arbitrarily small constant the discrete diffusion waves can be constructed from the self similar solutions of the heat equation and the burgers equation through an averaging process

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

estimates of the extremal solution for the bilaplacian with general nonlinearity let 0 denote the supremum possible value of such that 2u f u inb1 u u n 0on b1 has a classical solution where b1 is the unit ball in rn n is the exterior unit normal vector and f c1 r is nondecreasing and satisfies f 0 0 and f t t as t for this problem possesses a weak solution u the so called extremal solution we establish the regularity of this extremal solution for n 10 for n 11 we establish that limr 0 rn 82 u r limr 0 rn 102u r 0 for n 19 and limr 0 rn 92 u r limr 0 rn 112u r 0 for n 20 our regularity results do not depend on the specific nonlinearity f