nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

an example of blow up for the complex kdv equation and existence beyond the blow up the kdv equation with smooth complex initial data that blows up in a finite amount of time is studied first examples are given using the elliptic solutions of airault mckean and moser second the kdv flow of spectral coordinates is described showing that blow up occurs when eigenvalues run to infinity in finite time but eigenvalue collisions are harmless and do not cause a singularity eigenvalue collisions and blow up are studied numerically finally the solutions are continued through the singularity by continuing the coordinate flows on the appropriate riemann surface

the quasi stationary maxwell equations as singular limit of the complete equations the quasi linear case abstract the quasi stationary maxwell equations are considered as the time singular limit of the complete equations at the vanishing of the dielectric constant uniformly stable solutions of the complete equations are constructed and their convergence to a solution of the quasi stationary equations is proved and estimated

hyperbolic parabolic singular perturbation for quasilinear equations of kirchhoff type abstract we consider a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtain parameter dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of kirchhoff type and the corresponding quasilinear parabolic equation for this purpose we show time decay estimates for hyperbolic parabolic singular perturbation problem for linear equations with a time dependent coefficient

global existence of null form wave equations on small asymptotically euclidean manifolds abstract we prove the global existence of the small solutions to the cauchy problem for quasilinear wave equations satisfying the null condition on r 3 g where the metric g is a small perturbation of the flat metric and approaches the euclidean metric like 1 x 2 2 with 1 global and almost global existence for systems without the null condition are also discussed for certain small time dependent perturbations of the flat metric in appendix a

long time asymptotic expansion for the damped semilinear wave equation the first initial boundary value problem is considered for the damped semilinear wave equation with the quadratic nonlinearity for small initial data and homogeneous boundary conditions its solution is constructed in the form of a series in the eigenfunctions of the laplace operator the long time asymptotic expansion is obtained which shows the nonlinear effects of amplitude and frequency multiplication the same results hold for the admissible initial data that are not small

the cauchy problem for the ishimori equations abstract in 1984 y ishimori derived a nonlinear wave equation which is a two dimensional analogue of the classical isotropic spin chain in ferromagnetism in this paper we show that the cauchy problem associated to this equation is well posed in sobolev spaces when initial spins are almost parallel

on priors and bayes factors abstract by analogy to the authors measure of the influence of observations on bayes factors a new measure is proposed to determine the influence of a prior distribution on a bayes factor this is developed for linear models the method is illustrated for a consumption model using houthakker s international income and consumption data

bayesian regression diagnostics with applications to international consumption and income data abstract three variants of friedman s consumption model are analyzed using houthakker s international consumption and income data and the bayesian regression analysis program brap the analysis is focussed on evaluating friedman s proportionality hypothesis various diagnostic procedures are discussed and applied using diffuse and informative prior distributions of the three models considered the logit transformation model is deemed most appropriate and results based on it suggest a departure from the proportionality hypothesis particularlyat low levels of income this result is at variance with houthakker s earlier conclusion that these data are in accord with the proportionality hypothesis

hyperbolic to parabolic relaxation theory for quasilinear first order systems abstract in this paper we study the limiting behavior of nonhomogeneous hyperbolic systems of balance laws when the relaxed equilibria are described by means of systems of parabolic type in particular we obtain a complete theory for the 2 2 systems of genuinely nonlinear hyperbolic balance laws in 1 d with a strong dissipative term a different method which combines the div curl lemma with accretive operators is then applied to study the limiting profiles in the case of nonhomogeneous isentropic gas dynamics we also investigate relaxation results for some 2 d cases which include the cattaneo model for nonlinear heat conduction and the compressible euler flow moreover convergence result is also obtained for general semilinear systems in 1 d

analysis of two parareal algorithms for time periodic problems the parareal algorithm which permits us to solve evolution problems in a time parallel fashion has created a lot of attention over the past decade the algorithm has its roots in the multiple shooting method for boundary value problems which in the parareal algorithm is applied to initial value problems with a particular coarse approximation of the jacobian matrix it is therefore of interest to formulate parareal type algorithms for time periodic problems which also couple the end of the time interval with the beginning and to analyze their performance in this context we present and analyze two parareal algorithms for time periodic problems one with a periodic coarse problem and one with a nonperiodic coarse problem an interesting advantage of the algorithm with the nonperiodic coarse problem is that no time periodic problems need to be solved during the iteration since on the time subdomains the problems are not time periodic either we prove for both linear and nonlinear problems convergence

time asymptotics for solutions of vlasov poisson equation in a circle we prove that there exists a class of solutions of the nonlinear vlasov poisson equation vpe on a circle that converges weakly as t to a stationary homogeneous solution of vpe this behavior is called in the linear case landau damping the result is obtained by constructing a suitable scattering problem for the solutions of the vlasov poisson problem a consequence of this result is that a class of stationary solutions of the vlasov poisson equation is unstable in a weak topology

final state problem for korteweg de vries type equations we study the final state problem for the korteweg de vries type equations ut 1 x 1ux u2ux t x r r u t fs t l2 0 as t where r the function fs t we call a final state defined by the final data u we show that there does not exist a nontrivial solution of this equation in the case of fs t u t u where u t is the free evolution group of this equation we construct the modified wave operator for the korteweg de vries type equations under the conditions that the final data u arc real valued functions and the fourier transform u vanishes at the origin

fractional bayes factors for model comparison bayesian comparison of models is achieved simply by calculation of posterior probabilities of the models themselves however there are difficulties with this approach when prior information about the parameters of the various models is weak partial bayes factors offer a resoiution of the problem by setting aside part of the data as a training sampie the training sampie is used to obtain an initiai informative posterior distribution of the parameters in each model model comparison is then based on a bayes factor calculated from the remaining data properties of partial bayes factors are discussed particularly in the context of weak prior information and they are found to have advantages over other proposed methods of model comparison a new variant of the partial bayes factor the fractional bayes factor is advocated on grounds of consistency simplicity robustness and coherence

the global cauchy problem for the critical non linear wave equation abstract we study the global cauchy problem for the non linear wave equation p 1 0 for the critical value p n 2 n 2 in space dimension n 3 we identify a weak space time integrability property stip of the solutions and prove that it is sufficient to ensure the uniqueness of weak solutions the global existence of finite energy solutions with the naturally associated stip and the global existence of regular solutions with some n dependent restrictions on the regularity for spherically symmetric solutions we prove that the previous crucial stip follows from the morawetz inequality actually in a much stronger form than necessary thereby proving that all the previous results hold in the spherically symmetric case

long time existence and singular perturbation results for quasilinear hyperbolic equations with small parameter and dissipation term iii on considere des solutions lisses du probleme aux valeurs initiales hyperbolique quasilineaire eu tt u t ij 1 n a ij i j u 0 u 0 u 0 u t 0 u 1 ou e 0 0 u u x t r x r n on etudie l existence de solutions globales correspondant a des donnees arbitrairement grandes initiales u 0 et u 1 et a leur convergence quand e 0 vers la solution du probleme parabolique correspondant

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

global solutions of nonlinear wave equations in time dependent inhomogeneous media we consider the problem of small data global existence for a class of semilinear wave equations with null condition on a lorentzian background mathbb r 3 1 g with a time dependent metric g coinciding with the minkowski metric outside the cylinder t x x leq r we show that the small data global existence result can be reduced to two integrated local energy estimates and demonstrate that these estimates work in the particular case when g is merely c1 close to the minkowski metric one of the novel aspects of this work is that it applies to equations on backgrounds which do not settle to any particular stationary metric

numerical approximation of a periodic linear parabolic problem we approximate a periodic linear parabolic problem using a galerkin method for the space variable and a spectral method for the time variable some optimal error estimates are derived

approximation of hopf bifurcation we make several assumptions on a nonlinear evolution problem ensuring the existence of a hopf bifurcation under a fairly general approximation condition we define a discrete problem which retains the bifurcation property and we prove an error estimate between the branches of exact and approximate periodic solutions

measuring the effect of observations on bayes factors summary in this paper we consider a measure of the effect of single observations on a logarithmic bayes factor defined via the difference in the logarithms of the bayes factors conditional first on all the data and then omitting an observation the measure is related to the conditional predictive ordinate the form of the measure and examples of its use are presented for a variety of situations normal samples linear models log linear models and the checking of distributional assumptions

constructive error estimates for full discrete approximation of periodic solution for heat equation abstract we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation with time periodic condition our numerical scheme is based on the finite element semidiscretization in space direction combined with an interpolation in time using the fundamental matrix for the semidiscretized problem we derive the optimal order h 1 and l 2 error estimates that are critical in the numerical verification method of exact solutions for nonlinear parabolic equations several numerical examples that confirm the optimal rate of convergence are presented

decay and scattering of small solutions of a generalized boussinesq equation abstract we study the long time behavior of small solutions of the initial value problem for a generalized boussinesq equation we obtain a lower bound for the degrees of nonlinearity which allows us to establish a nonlinear scattering result for small perturbations that is the small solutions of the nonlinear problem behave asymptotically like the solution of the associated linear problem under certain hypotheses we can construct a scattering operator for the boussinesq equation which carries a neighborhood of 0 in the energy space x into x

lower bounds for the life span of solutions of nonlinear wave equations in three dimensions abstract the paper deals with strict solutions u x t u x1 x2 x3 t of an equation formula see text where du is the set of four first derivatives of u for given initial values u x 0 ef x ut x 0 eg x the life span t e is defined as the supremum of all t to which the local solution can be extended for all x blowup in finite time corresponds to t e examples show that this can occur for arbitrarily small e on the other hand t e must at least be very large for small e by assuming that aik f g unk c that aik 0 0 and that f g have compact support it is shown that formula see text for every n this result had been established previously only for n 4

global existence of small solutions to nonlinear evolution equations abstract we study the global existence and asymptotic behaviour of small solutions of a large class of nonlinear partial differential equations if the nonlinear terms are of high degree the solutions will be asymptotic to solutions of the linear equation