the calderon problem with partial data in this paper we improve an earlier result by bukhgeim and uhlmann 1 by showing that in dimension n 3 the knowledge of the cauchy data for the schr nodinger equation measured on possibly very small subsets of the boundary determines uniquely the potential we follow the general strategy of 1 but use a richer set of solutions to the dirichlet problem this implies a similar result for the problem of electrical impedance tomography which consists in determining the conductivity of a body by making voltage and current measurements at the boundary

stability analysis for pitchfork bifurcations of solitary waves in generalized nonlinear schrodinger equations abstract linear stability of both sign definite positive and sign indefinite solitary waves near pitchfork bifurcations is analyzed for the generalized nonlinear schrodinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions bifurcations of linear stability eigenvalues associated with pitchfork bifurcations are analytically calculated based on these eigenvalue bifurcation formulae linear stability of solitary waves near pitchfork bifurcations is then determined it is shown that the base solution branch switches stability at the bifurcation point in addition the two bifurcated solution branches and the base branch have the opposite same stability when their power slopes have the same opposite sign furthermore the stability of these solution branches can be determined almost exclusively from their power diagram especially for positive solitary waves these stability results are also compared with the hamiltonian krein index theory and they are shown to be consistent with each other lastly various numerical examples are presented and the numerical results confirm the analytical predictions both qualitatively and quantitatively

analysis of the linearization around a critical point of an infinite dimensional hamiltonian system partial differential equations that conserve energy can often be written as infinite dimensional hamiltonian systems of the following general form du dt je u where j x x is a symplectic matrix i e jj 1 and e x r is a c 2 functional defined on some hilbert space x

count of eigenvalues in the generalized eigenvalue problem we study isolated and embedded eigenvalues in the generalized eigenvalue problem defined by two self adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues the generalized eigenvalue problem determines the spectral stability of nonlinear waves in infinite dimensional hamiltonian systems the theory is based on pontryagin s invariant subspace theorem and extends beyond the scope of earlier papers of pontryagin krein grillakis and others our main results are i the number of unstable and potentially unstable eigenvalues equals the number of negative eigenvalues of the self adjoint operators ii the total number of isolated eigenvalues of the generalized eigenvalue problem is bounded from above by the total number of isolated eigenvalues of the self adjoint operators and iii the quadratic forms defined by the two self adjoint operators are strictly positive on the subspace related to the continuous spectrum of the generalized eigenvalue problem applicatio

stability of the bloch wall via the bogomolnyi decomposition in elliptic coordinates we consider the one dimensional anisotropic xy model in the continuum limit stability analysis of its bloch wall solution is hindered by the non diagonality of the associated linearized operator and the hessian of energy we circumvent this difficulty by showing that the energy admits a bogomolnyi bound in elliptic coordinates and that the bloch wall saturates it that is the bloch wall renders the energy minimum our analysis provides a simple but nontrivial application of the bps bogomolnyi prasad sommerfield construction in one dimension where its use is often believed to be limited to reproducing results obtainable by other means

on the markov sequence problem for jacobi polynomials abstract we give a simple and entirely elementary proof of gasper s theorem on the markov sequence problem for jacobi polynomials it is based on the spectral analysis of an operator that arises in the study of a probabilistic model of colliding molecules introduced by marc kac and the methods developed here yield new estimates relevant to the collision model

graphical krein signature theory and evans krein functions two concepts evidently very different in nature have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory i the krein signature of an eigenvalue a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right half plane leading to dynamical instability under perturbation of the system and ii the evans function an analytic function detecting the location of eigenvalues one might expect these two concepts to be related but unfortunately examples demonstrate that there is no way in general to deduce the krein signature of an eigenvalue from the evans function for example by studying derivatives of the latter the purpose of this paper is to recall and popularize a simple graphical interpretation of the krein signature well known in the spectral theory of polynomial operator pencils once esta

cercignani s conjecture is sometimes true and always almost true we establish several new functional inequalities comparing boltzmann s entropy production functional with the relative h functional first we prove a longstanding conjecture by cercignani under the nonphysical assumption that the boltzmann collision kernel is superquadratic at infinity the proof rests on the method introduced in 39 combined with a novel use of the blachman stam inequality if the superquadraticity assumption is not satisfied then it is known that cercignani s conjecture is not true however we establish a slightly weakened version of it for all physically relevant collision kernels thus extending previous results from 39 finally we consider the entropy entropy production version of kac s spectral gap problem and obtain estimates about the dependence of the constants with respect to the dimension the first two results are sharp in some sense and the third one is likely to be too they contain all previously known entropy estimates as particular cases this gives a first coherent picture of the study of entropy production according to a program started by carlen and carvalho 12 ten years ago these entropy inequalities are one step in our study of the trend to equilibrium for the boltzmann equation in both its spatially homogeneous and spatially inhomogeneous versions

spectral stability and time evolution of n solitons in the kdv hierarchy this paper concerns spectral stability and time evolution of n solitons in the korteweg de vries kdv hierarchy with mixed commuting time flows spectral stability problem is analysed by using a pair of self adjoint operators with finite numbers of negative eigenvalues we show that the absence of unstable eigenvalues in the stability problem is related to the absence of negative eigenvalues of these operators in the constrained function spaces time evolution of n solitons is uniquely characterized from the inverse scattering transform technique

on the classification of the spectrally stable standing waves of the hartree problem abstract we consider the fractional hartree model with general power non linearity and arbitrary spatial dimension we construct variationally the normalized solutions for the corresponding choquard pekar model in particular a number of key properties like smoothness and bell shapedness are established as a consequence of the construction we show that these solitons are spectrally stable as solutions to the time dependent hartree model in addition we analyze the spectral stability of the moroz van schaftingen solitons of the classical hartree problem in any dimensions and power non linearity a full classification is obtained the main conclusion of which is that only and exactly the normalized solutions which exist only in a portion of the range are spectrally stable

on the rate of entropy production for the boltzmann equation we show that there exists a wide class of distribution functions with moments of any order as close to their equilibrium values as we like which can provide an abnormally low rate of entropy production the result is valid for the boltzmann equation with any cross section v satisfying a mild restriction the functions are constructed in an explicit form and we discuss some applications of our results

an inverse problem for the dynamical lame system with two sets of boundary data we prove uniqueness and a holder type stability of reconstruction of all three time independent elastic parameters in the dynamical isotropic system of elasticity from two special sets of boundary measurements in proofs we use carleman type estimates in sobolev spaces of negative order 2003 wiley periodicals inc

spectral stability analysis for special solutions of second order in time pdes the higher dimensional case abstract we develop a general theory to treat the linear stability of certain special solutions of second order in time evolutionary pde we apply these results to standing waves of the following problems the klein gordon equation for which we consider both ground states and certain excited states the klein gordon zakharov system and the beam equation we also discuss applications to excited states for the klein gordon model as well as multidimensional traveling waves not necessarily homoclinic to zero for general second order equations of this type in all cases our abstract results provide a complete characterization of the linear stability of such solutions

eigenvalues of zero energy in the linearized nls problem we study a pair of neutrally stable eigenvalues of zero energy in the linearized nls equation we prove that the pair of isolated eigenvalues where each eigenvalue has geometric multiplicity one and algebraic multiplicity n is associated with 2p negative eigenvalues of the energy operator where p n 2 if n is even and p n 1 2 or p n 1 2 if n is odd when the potential of the linearized nls problem is perturbed due to parameter continuations we compute the exact number of unstable eigenvalues that bifurcate from the neutrally stable eigenvalues of zero energy

insensitizing controls for a class of quasilinear parabolic equations abstract this paper is addressed to showing the existence of insensitizing controls for a class of quasilinear parabolic equations with homogeneous dirichlet boundary conditions as usual this insensitizing problem is reduced to a nonstandard null controllability problem of some nonlinear cascade system governed by a quasilinear parabolic equation and a linear parabolic equation nevertheless in order to solve the later quasilinear controllability problem by the fixed point technique we need to establish the null controllability of the linearized cascade parabolic system in the framework of classical solutions the key point is to find the desired control function in a holder space for given data with certain regularities

on nonlocal gross pitaevskii equations with periodic potentials the gross pitaevskii equation is a widely used model in physics in particular in the context of bose einstein condensates however it only takes into account local interactions between particles this paper demonstrates the validity of using a nonlocal formulation as a generalization of the local model in particular the paper demonstrates that the solution of the nonlocal model approaches in norm the solution of the local model as the nonlocal model approaches the local model the nonlocality and potential used for the gross pitaevskii equation are quite general thus this paper shows that one can easily add nonlocal effects to interesting classes of bose einstein condensate models based on a particular choice of potential for the nonlocal gross pitaevskii equation we establish the orbital stability of a class of parameter dependent solutions to the nonlocal problem for certain parameter regimes numerical results corroborate the analytical stability results and lead to predictions about the stability of the class of solutions for parameter values outside of the purview of the theory established in this paper

global carleman inequalities for parabolic systems and applications to controllability this paper has been conceived as an overview on the controllability properties of some relevant linear and nonlinear parabolic systems specifically we deal with the null controllability and the exact controllability to the trajectories we try to explain the role played by the observability inequalities in this context and the need of global carleman estimates we also recall the main ideas used to overcome the difficulties motivated by nonlinearities first we considered the classical heat equation with dirichlet conditions and distributed controls then we analyze recent extensions to other linear and semilinear parabolic systems and or boundary controls finally we review the controllability properties for the stokes and navier stokes equations that are known to date in this context we have paid special attention to obtaining the necessary carleman estimates some open questions are mentioned throughout the paper we hope that this unified presentation will be useful for those researchers interested in the field

on the spectra of periodic waves for infinite dimensional hamiltonian systems abstract we consider the problem of determining the spectrum for the linearization of an infinite dimensional hamiltonian system about a spatially periodic traveling wave by using a bloch wave decomposition we recast the problem as determining the point spectra for a family of operators j l where j is skew symmetric with bounded inverse and l is symmetric with compact inverse our main result relates the number of unstable eigenvalues of the operator j l to the number of negative eigenvalues of the symmetric operator l the compactness of the resolvent operators allows us to greatly simplify the proofs as compared to those where similar results are obtained for linearizations about localized waves the theoretical results are general and apply to a larger class of problems than those considered herein the theory is applied to a study of the spectra associated with periodic and quasi periodic solutions to the nonlinear schrodinger equation as well as periodic solutions to the generalized korteweg de vries equation with power nonlinearity

carleman estimates for the one dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem abstract we study the observability and some of its consequences controllability identification of diffusion coefficients for one dimensional heat equations with discontinuous coefficients piecewise c 1 the observability for a linear equation is obtained by a carleman type estimate this kind of observability inequality yields controllability results for a semi linear equation as well as a stability result for the identification of the diffusion coefficient

counting eigenvalues via the krein signature in infinite dimensional hamiltonian systems spectra of nonlinear waves in infinite dimensional hamiltonian systems are investigated we establish a connection via the krein signature between the number of negative directions of the second variation of the energy and the number of potentially unstable eigenvalues of the linearization about a nonlinear wave we apply our results to determine the effect of symmetry breaking on the spectral stability of nonlinear waves in weakly coupled nonlinear schrodinger equations

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

spectral gap for kac s model of boltzmann equation we consider a random walk on s n 1 the standard sphere of dimension n 1 generated by random rotations on randomly selected coordinate planes i j with 1 i j n this dynamic was used by marc kac as a model for the spatially homogeneous boltzmann equation we prove that the spectral gap on s n 1 is n 1 up to a constant independent of n

carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations in a bounded domain of r n 1 n geq 2 we consider a second order elliptic operator a d x 0 2 nabla x cdot c x nabla x where the scalar coefficient c x is piecewise smooth yet discontinuous across a smooth interface s we prove a local carleman estimate for a in the nhd of any point of the interface the observation region can be chosen independently of the sign of the jump of the coefficient c at the considered point the derivation of this estimate relies on the separation of the problem into three microlocal regions and the calderon projector technique following the method of lebeau and robbiano cite lr 95 we then prove the null controllability for the linear parabolic initial problem with dirichlet boundary conditions associated to the operator d t nabla x cdot c x nabla x

bifurcations from the endpoints of the essential spectrum in the linearized nonlinear schrodinger problem we study bifurcations of eigenvalues from the endpoints of the essential spectrum in the linearized nonlinear schrodinger problem in three dimensions we show that a resonance and an eigenvalue of positive energy at the endpoint may bifurcate only to a real eigenvalue of positive energy while an eigenvalue of negative energy at the endpoint may also bifurcate to complex eigenvalues

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

local and global carleman estimates for parabolic operators with coefficients with jumps at interfaces in 0 t open subset of n n 2 we consider a parabolic operator p t x t x x where the scalar coefficient t x is piecewise smooth in space yet discontinuous across a smooth interface s we prove a global in time local in space carleman estimate for p in the neighborhood of any point of the interface the observation region can be chosen independently of the sign of the jump of the coefficient at the considered point the derivation of this estimate relies on the separation of the problem into three microlocal regions related to high and low tangential frequencies at the interface in the high frequency regime we use calderon projectors in the low frequency regime we follow a more classical approach because of the parabolic nature of the problem we need to introduce weyl hormander anisotropic metrics symbol classes and pseudo differential operators each frequency regime and the associated technique require a different calculus a global in time and space carleman estimate on 0 t m m a manifold is also derived from the local result

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

variational necessary and sufficient stability conditions for inviscid shear flow a necessary and sufficient condition for linear stability of inviscid parallel shear flow is formulated by developing a novel variational principle where the velocity profile is assumed to be monotonic and analytic it is shown that unstable eigenvalues of rayleigh s equation which is a non self adjoint eigenvalue problem can be associated with positive eigenvalues of a certain self adjoint operator the stability is therefore determined by maximizing a quadratic form which is theoretically and numerically more tractable than directly solving rayleigh s equation this variational stability criterion is based on the understanding of kre n signature for continuous spectra and is applicable to other stability problems of infinite dimensional hamiltonian systems

carleman estimates for stratified media abstract we consider anisotropic elliptic and parabolic operators in a bounded stratified media in r n characterized by discontinuities of the coefficients in one direction the surfaces of discontinuities cross the boundary of the domain we prove carleman estimates for these operators with an arbitrary observation region

nonlinear stability of gardner breathers abstract we show that breather solutions of the gardner equation a natural generalization of the kdv and mkdv equations are h 2 r stable through a variational approach we characterize gardner breathers as minimizers of a new lyapunov functional and we study the associated spectral problem through i the analysis of the spectrum of explicit linear systems spectral stability and i i controlling degenerated directions by using low regularity conservation laws

spectra of positive and negative energies in the linearized nls problem we study the spectrum of the linearized nls equation in three dimensions in association with the energy spectrum we prove that unstable eigenvalues of the linearized nls problem are related to negative eigenvalues of the energy spectrum while neutrally stable eigenvalues may have both positive and negative energies the nonsingular part of the neutrally stable essential spectrum is always related to the positive energy spectrum we derive bounds on the number of unstable eigenvalues of the linearized nls problem and study bifurcations of embedded eigenvalues of positive and negative energies we develop the l 2 scattering theory for the linearized nls operators and recover results of grillakis 5 with a fermi golden rule c

carleman estimates and controllability results for the one dimensional heat equation with bv coefficients abstract we derive global carleman estimates for one dimensional linear parabolic equations t x c x with a coefficient of bounded variations these estimates are obtained by approximating c by piecewise constant coefficients c e and passing to the limit in the carleman estimates associated to the operators defined with c e such estimates yields observability inequalities for the considered linear parabolic equation which in turn yield controllability results for classes of semilinear equations

periodic finite genus solutions of the kdv equation are orbitally stable abstract the stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade the kdv equation is known to have large families of periodic solutions that are parameterized by hyperelliptic riemann surfaces they are generalizations of the famous multi soliton solutions we show that all such periodic solutions are orbitally stable with respect to subharmonic perturbations perturbations that are periodic with period equal to an integer multiple of the period of the underlying solution

equilibration in the kac model using the gtw metric d_2 we use the fourier based gabetta toscani wennberg metric d 2 to study the rate of convergence to equilibrium for the kac model in 1 dimension we take the initial velocity distribution of the particles to be a borel probability measure mu on mathbb r n that is symmetric in all its variables has mean vec 0 and finite second moment let mu t dv denote the kac evolved distribution at time t and let r mu be the angular average of mu we give an upper bound to d 2 mu t r mu of the form min left b e frac 4 lambda 1 n 3 t d 2 mu r mu right where lambda 1 frac n 2 2 n 1 is the gap of the kac model in l 2 and b depends only on the second moment of mu we also construct a family of schwartz probability densities f 0 n mathbb r n rightarrow mathbb r with finite second moments that shows practically no decrease in d 2 f 0 t r f 0 for time at least frac 1 2 lambda with lambda the rate of the kac operator we also present a propagation of chaos result for the partially thermostated kac model in tossounian and vaidyanathan j math phys 56 8 083301 2015

metrics for probability distributions and the trend to equilibrium for solutions of the boltzmann equation this paper deals with the trend to equilibrium of solutions to the spacehomogeneous boltzmann equation for maxwellian molecules with angular cutoff as well as with infinite range forces the solutions are considered as densities of probability distributions the tanaka functional is a metric for the space of probability distributions which has previously been used in connection with the boltzmann equation our main result is that if the initial distribution possesses moments of order 2 e then the convergence to equilibrium in his metric is exponential in time in the proof we study the relation between several metrics for spaces of probability distributions and relate this to the boltzmann equation by proving that the fourier transformed solutions are at least as regular as the fourier transform of the initial data this is also used to prove that even if the initial data only possess a second moment then v r f v t v 2 dv 0 asr and this convergence is uniform in time

inertia law for spectral stability of solitary waves in coupled nonlinear schrodinger equations spectral stability analysis for solitary waves is developed in the context of the hamiltonian system of coupled nonlinear schrodinger equations the linear eigen value problem for a non self adjoint operator is studied with two self adjoint matrix schrodinger operators sharp bounds on the number and type of unstable eigen values in the spectral problem are found from the inertia law for quadratic forms associated with the two self adjoint operators symmetry breaking stability analysis is also developed with the same method

null controllability of a system of linear thermoelasticity we consider a linear system of thermoelasticity in a compact c infin n dimensional connected riemannian manifold this system consists of a wave equation coupled to a heat equation when the boundary of the manifold is non empty dirichlet boundary conditions are considered we study the controllability properties of this system when the control acts in the hyperbolic equation and not in the parabolic one and has its support restricted to an open subset of the manifold we show that if the control time and the support of the control satisfy the geometric control condition for the wave equation this system of thermoelasticity is null controllable more precisely any finite energy solution can be driven to zero at the control time an analogous result is proved when the control acts on the parabolic equation finally when the manifold has no boundary the null controllability of the linear system of three dimensional thermoelastic ity is proved

linearized instability for nonlinear schrodinger and klein gordon equations in this paper i am introducing a new technique for proving instability of bound states for hamiltonian systems there are already two disparate types of instability results in the literature the approach developed by strauss shatah 20 gave an instability criterion coming from the variational structure of the problem on the other hand jones approach 11 produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques it turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results finally in order to demonstrate how this method works i apply the instability criterion in some specific examples