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

strichartz estimates for the wave and schrodinger equations with the inverse square potential abstract we prove spacetime weighted l 2 estimates for the schrodinger and wave equation with an inverse square potential we then deduce strichartz estimates for these equations

global existence for the one dimensional second order semilinear hyperbolic equations abstract the aim of the paper is to study necessary and sufficient conditions for the existence of the global solution of the one dimensional semilinear equation appearing in the boundary value problems of gas dynamics we investigate the cauchy problem for such equation in the domain where the operator is weakly hyperbolic we obtain the necessary condition for the existence of the self similar solutions for the semilinear gellerstedt type equation the approach used in the paper is based on the fundamental solution of the linear gellerstedt operator and the l p l q estimates

normal forms and quadratic nonlinear klein gordon equations on construit une transformation qui change l equation a non linearite cubique que l on sait resoudre a 3 1 dimensions on demontre que la solution du probleme original a le meme comportement asymptotique que l equation de klein gordon non lineaire

nonlinear scattering theory at low energy sequel abstract we extend the nonlinear scattering theory and apply it to schrodinger and klein gordon equations with various kinds of nonlinear terms including some nonlocal ones

nonlinear small data scattering for the wave and klein gordon equation on etudie l operateur diffusion qui appartient a la paire d equations u tt au f u 0 et u tt au 0 a represente l operateur j 1 n 2 x j 2 m 2 m r et f u u 1 u ou r 1

sharp threshold of global existence and blow up of the combined nonlinear klein gordon equation abstract in this paper we show the global well posedness and blow up result of the solutions with the energy below the threshold for the combined nonlinear klein gordon equation u t t u u u p 1 u u q 1 u d 3 in the energy space h 1 r d l 2 r d where 1 4 d q p 1 4 d 2 we give a threshold of blow up and global well posedness using a modified variational approach in the spirit of ibrahim et al 2011 7

regularity of solutions to nonlinear dispersive equations on considere le probleme de cauchy pour les equations dispersives non lineaires de la forme t u ip d u f u x r n t u x 0 u 0 x

lp lp estimates for the schrodinger equation on the line and inverse scattering for the nonlinear schrodinger equation with a potential abstract in this paper i prove a l p l p estimate for the solutions to the one dimensional schrodinger equation with a potential in l 1 where in the generic case 3 2 and in the exceptional case i e when there is a half bound state of zero energy 5 2 i use this estimate to construct the scattering operator for the nonlinear schrodinger equation with a potential i prove moreover that the low energy limit of the scattering operator uniquely determines the potential and the coupling constant of the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonlinearity

global existence for the dissipative wave equations with space time dependent potential abstract this research paper aims at studying the long time behavior of solutions for wave equations with focusing nonlinearity we consider p c r n p n 2 n 2 and separable potential a t x x t with x 1 x and t 1 t satisfying conditions 3 4 which are given below the main result is that if the exponent p lies between p c r 1 4 1 2 n 1 2 p n 2 n 2 the solutions are global for sufficiently small data also the precise decay estimates for the energy l 2 and l p 1 norms of solutions in the region of global solutions have been found

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

modified scattering for the critical nonlinear schrodinger equation abstract we consider the nonlinear schrodinger equation i u t u u 2 n u in all dimensions n 1 where c and 0 we construct a class of initial values for which the corresponding solution is global and decays as t like t n 2 if 0 and like t log t n 2 if 0 moreover we give an asymptotic expansion of those solutions as t we construct solutions that do not vanish so as to avoid any issue related to the lack of regularity of the nonlinearity at u 0 to study the asymptotic behavior we apply the pseudo conformal transformation and estimate the solutions by allowing a certain growth of the sobolev norms which depends on the order of regularity through a cascade of exponents

analyticity of the scattering operator for the nonlinear klein gordon equation with cubic nonlinearity the wave and scattering operators for the equation left square m 2 right varphi lambda varphi 2 0 withm 0 and 0 on four dimensional minkowski space are analytic on the space of finite energy cauchy data i e l21 r3 l2 r3

intrusive galerkin methods with upwinding for uncertain nonlinear hyperbolic systems this paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws we consider problems with parametric uncertainty in initial conditions and model coefficients whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables the stochastic spectral method relies on multi resolution schemes where the stochastic domain is discretized using tensor product stochastic elements supporting local polynomial bases a galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution hyperbolicity of the resulting galerkin system is analyzed a finite volume scheme with a roe type solver is used for discretization of the spatial and time variables an original technique is introduced for the fast evaluation of approximate upwind matrices which is particularly well adapted to local polynomial bases efficiency and robustness of the overall method are assessed on the burgers and euler equations with shocks

dynamical systems that perform the singular value decomposition abstract a dynamical system is presented whose solutions are shown to converge exponentially to the singular value decomposition of a given matrix provided that its singular values are distinct this system has potential application to certain problems in adaptive control and adaptive filtering it is one of several techniques that employ dynamical systems to perform tasks done conventionally with the techniques of numerical linear algebra

a note on the 2d generalized zakharov kuznetsov equation local global and scattering results abstract we consider the generalized two dimensional zakharov kuznetsov equation u t x u x u k 1 0 where k 3 is an integer number for k 8 we prove local well posedness in the l 2 based sobolev spaces h s r 2 where s is greater than the critical scaling index s k 1 2 k for k 3 we also establish a sharp criteria to obtain global h 1 r 2 solutions a nonlinear scattering result in h 1 r 2 is also established assuming the initial data is small and belongs to a suitable lebesgue space

stochastic model reduction for chaos representations abstract this paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic galerkin schemes for the solution of partial differential equations with stochastic coefficients in particular an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation spde in the hilbert space spanned by wiener chaos a description of the stochastic aspect of the problem on two well separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale with such algorithms at hand the solution of spde s becomes both computationally manageable and efficient moreover a solid foundation is thus provided for the adaptive error control in stochastic galerkin procedures different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics

nonlinear scattering with nonlocal interaction we consider the scattering problem for the hartree type equation in n withn 2 i frac partial u partial t frac 1 2 delta u v u 2 u where v x sum limits j 1 2 lambda j x gamma j lambda 1 lambda 2 ne 0 0 lambda j in mathbb r gamma j 0 and denotes the convolution in n we prove the existence of wave operators inh0 k l2 n x k l2 n for any positive integerk under the assumption 1 1 2 2 this is an optimal result in the sense that the existence of wave operators breaks down if min 1 2 1 the case where 1 1 2 2 is also treated according to the sign of 2

bicyclic signed graphs with minimal and second minimal energy abstract let s g be a signed graph of order n and size m where g is its underlying graph and is the sign function a connected signed graph s is said to be bicyclic if m n 1 in this paper we provide characterizations of bicyclic signed graphs with minimal and second minimal energy we notice that there are gaps in the proofs of the main results in j zhang h kan 2014 32 and j zhang b zhou 2005 31 which deal with minimal energy of bicyclic graphs and we plug these gaps as a consequence of the results obtained for bicyclic signed graphs

combined effects of two nonlinearities in lifespan of small solutions to semi linear wave equations this paper investigates the combined effects of two distinctive power type nonlinear terms with parameters p q 1 in the lifespan of small solutions to semi linear wave equations we determine the full region of p q to admit global existence of small solutions at least for spatial dimensions n 2 3 moreover for many p q when there is no global existence we obtain sharp lower bound of the lifespan which is of the same order as the upper bound of the lifespan

stochastic sensitivity a computable lagrangian uncertainty measure for unsteady flows uncertainties in velocity data are often ignored when computing lagrangian particle trajectories of fluids modeling these as noise in the velocity field leads to a random deviation from each traje

dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty abstract we estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations within the context of dynamically orthogonal do field equations transient nonlinear dynamics irregular data and non stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation to efficiently quantify uncertainties in such systems it is essential to vary the dimensionality of the stochastic subspace with time an objective here is to provide criteria to do so working directly with the original equations of the dynamical system under study and its do representation we first analyze the scaling of the computational cost of these do equations with the stochastic dimensionality and show that unlike many other stochastic methods the do equations do not suffer from the curse of dimensionality subsequently we present the new adaptive criteria for the variation of the stochastic dimensionality based on instantaneous i stability arguments and ii bayesian data updates we then illustrate the capabilities of the derived criteria to resolve the transient dynamics of two 2d stochastic fluid flows specifically a double gyre wind driven circulation and a lid driven cavity flow in a basin in these two applications we focus on the growth of uncertainty due to internal instabilities in deterministic flows we consider a range of flow conditions described by varied reynolds numbers and we study and compare the evolution of the uncertainty estimates under these varied conditions

global nonexistence results for a class of hyperbolic systems abstract our concern in this paper is to prove blow up results to the non autonomous nonlinear system of wave equations u t t u a t x v p v t t v b t x u q t 0 x r n in any space dimension we show that a curve f p q 0 depending on the space dimension on the exponents p q and on the behavior of the functions a t x and b t x exists such that all nontrivial solutions to the above system blow up in a finite time whenever f p q 0 our method of proof uses some estimates developed by galaktionov and pohozaev in 11 for a single non autonomous wave equation enabling us to obtain a system of ordinary differential inequalities from which the desired result is derived our result generalizes some important results such as the ones in del santo et al 1996 12 and galaktionov and pohozaev 2003 11 the advantage here is that our result applies to a wide variety of problems

on space time means and everywhere defined scattering operators for nonlinear klein gordon equations on demontre l existence d operateurs de diffusion definis partout pour certaines equations de klein gordon non lineaires

finite time blow up to critical semilinear wave equation outside the ball in 3 d abstract in this paper we study the initial boundary value problem of critical semilinear wave equation with small initial data in 3 d blow up result will be established no matter how small the data are we find a radial and positive solution of the corresponding linear problem then our blow up result follows from the idea of zhou and han 2014 what is more we get the upper bound of the lifespan estimate

error analysis of the dynamically orthogonal approximation of time dependent random pdes in this work we discuss the dynamically orthogonal do approximation of time dependent partial differential equations with random data the approximate solution is expanded at each time instant on a time dependent orthonormal basis in the physical domain with a fixed and small number of terms dynamic equations are written for the evolution of the basis as well as the evolution of the stochastic coefficients of the expansion we analyze the case of a linear parabolic equation with random data and derive a theoretical bound for the approximation error of the s terms do solution by the corresponding s terms best approximation i e the truncated s terms karhunen loeve expansion at each time instant the bound is applicable on the largest time interval in which the best s terms approximation is continuously time differentiable properties of the do approximations are analyzed on simple cases of deterministic equations with random initial data numerical tests are presented that confirm the theoret

projection like retractions on matrix manifolds this paper deals with constructing retractions a key step when applying optimization algorithms on matrix manifolds for submanifolds of euclidean spaces we show that the operation consisting of taking a tangent step in the embedding euclidean space followed by a projection onto the submanifold is a retraction we also show that the operation remains a retraction if the projection is generalized to a projection like procedure that consists of coming back to the submanifold along admissible directions and we give a sufficient condition on the admissible directions for the generated retraction to be second order this theory offers a framework in which previously proposed retractions can be analyzed as well as a toolbox for constructing new ones illustrations are given for projection like procedures on some specific manifolds for which we have an explicit easy to compute expression

finite time blowup of solutions to semilinear wave equation in an exterior domain abstract we consider the initial boundary value problem of semilinear wave equation with nonlinearity u p in exterior domain in r n n 3 especially the lifespan of blowup solutions with small initial data are studied the result gives upper bounds of lifespan which is essentially the same as the cauchy problem in r n at least in the case n 4 their estimates are sharp in view of the work by zha zhou 21 the idea of the proof is to use special solutions to linear wave equation with dirichlet boundary condition which are constructed via an argument based on wakasa yordanov 15

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

nonlinear scattering theory for a class of wave equations in hs abstract for the semi linear higher order wave equation and the nonlinear higher order schrodinger equation we show that the scattering operators map a band in h s into h s if the nonlinearities have sub critical powers in h s the smoothness of the scattering operators and the uniform boundedness of strong solutions for the defocusing nls equation are also shown provided that the nonlinearities have subcritical growth in h 1 moreover the spatial decaying behavior of solutions in energy space for the defocusing nls equation are obtained

lower bounds of asymptotics in time of solutions to nonlinear schrodinger equations in 3d abstract we study sharp time asymptotics of solutions around the final states of nonlinear schrodinger equations with quadratic nonlinearities in three space dimensions in particular we consider the large time asymptotic estimates of solutions from below

dynamically orthogonal numerical schemes for efficient stochastic advection and lagrangian transport quantifying the uncertainty of lagrangian motion can be performed by solving a large number of ordinary differential equations with random velocities or equivalently a stochastic transport partial differential equation pde for the ensemble of flow maps the dynamically orthogonal do decomposition is applied as an efficient dynamical model order reduction to solve for such stochastic advection and lagrangian transport its interpretation as the method that applies instantaneously the truncated svd on the matrix discretization of the original stochastic pde is used to obtain new numerical schemes fully linear explicit central advection schemes stabilized with numerical filters are selected to ensure efficiency accuracy stability and direct consistency between the original deterministic and stochastic do advections and flow maps various strategies are presented for selecting a time stepping that accounts for the curvature of the fixed rank manifold and the error related to closely singular coefficient matrices efficient schemes are developed to dynamically evolve the rank of the reduced solution and to ensure the orthogonality of the basis matrix while preserving its smooth evolution over time finally the new schemes are applied to quantify the uncertain lagrangian motions of a 2d double gyre flow with random frequency and of a stochastic flow past a cylinder

an eulerian approach for computing the finite time lyapunov exponent we propose efficient eulerian methods for approximating the finite time lyapunov exponent ftle the idea is to compute the related flow map using the level set method and the liouville equation there are several advantages of the proposed approach unlike the usual lagrangian type computations the resulting method requires the velocity field defined only at discrete locations no interpolation of the velocity field is needed also the method automatically stops a particle trajectory in the case when the ray hits the boundary of the computational domain the computational complexity of the algorithm is o dx d 1 with d the dimension of the physical space since there are the same number of mesh points in the x t space the computational complexity of the proposed eulerian approach is optimal in the sense that each grid point is visited for only o 1 time we also extend the algorithm to compute the ftle on a co dimension one manifold the resulting algorithm does not require computation on any local coordinate system and is simple to implement even for an evolving manifold

pollution release tied to invariant manifolds a case study for the coast of florida high resolution ocean velocity data has become readily available since the introduction of very high frequency vhf radar technology the vast amount of data generated so far however remains largely unused in environmental prediction in this paper we use vhf data of the florida coastline to locate lagrangian coherent structures lcs hidden in ocean surface currents such structures govern the spread of organic contaminants and passive drifters that stay confined to the ocean surface we use the lagrangian structures in a real time pollution release scheme that reduces the effect of industrial contamination on the coastal environment

new blow up results for nonlinear boundary value problems in exterior domains abstract we consider exterior boundary value problems with nontrivial boundary conditions including three types of semilinear equations heat equation wave equation and damped wave equation using a test function approach with a choice of a suitable family of test functions we obtain the critical exponents in the sense of fujita for the considered problems our obtained results close some open questions from the literature

critical criteria of fujita type for a system of inhomogeneous wave inequalities in exterior domains abstract we consider blow up results for a system of inhomogeneous wave inequalities in exterior domains we will handle three type boundary conditions dirichlet type neumann type and mixed boundary conditions we use a unified approach to show the optimal criteria of fujita type for each case our study yields naturally optimal nonexistence results for the corresponding stationary wave system and equation we provide many new results and close some open questions

critical blowup for systems of semilinear wave equations in low space dimensions abstract we consider the cauchy problem for systems of semilinear wave equations in two and three space dimensions with small initial data del santo et al geometric optics and related topics f colombini and n lerner eds progress in nonlinear differential equations and their applications vol 32 pp 117 140 birkhauser boston 1997 have studied the existence and nonexistence of global classical solutions of the cauchy problem except for the critical case in this paper we study the critical case and we show the nonexistence of global classical solutions and also give the upper bounds of the life span

lower bounds for the life span of solutions of nonlinear wave equations in higher dimensions when m 0 writing 1 as a quasilinear symmetric hyperbolic system it is known that a classical solution exists locally in time provided that f g are small and sufficiently smooth the time interval o t for which this classical solution exists depends on the size of the initial data more precisely if f and g go where fo go are fixed the largest time t t e for which the classical solution of 1 is assured by the local existence theorem is of order l for the one dimensional genuine nonliner wave equation i e m 0 n 1 in 1 and cy 1 in 2 lax l has shown that this estimate is sharp in this case when the genuine nonlinearity condition is violated i e ly 1 klainerman and majda 2 have shown that t e 0 l and that this estimate is also sharp for e sufficiently small in contrast to the one dimensional case john has shown that in higher dimensions the existence time of classical solutions can be significantly improved in fact he proved for the case m 0 y 1 that t e 0 l s for n 4 5 3 and recently t e u l 8 for

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

analyticity of the scattering operator for semilinear dispersive equations we present a general algorithm to show that a scattering operator associated to a semilinear dispersive equation is real analytic and to compute the coefficients of its taylor series at any point we illustrate this method in the case of the schrodinger equation with power like nonlinearity or with hartree type nonlinearity and in the case of the wave and klein gordon equations with power nonlinearity finally we discuss the link of this approach with inverse scattering and with complete integrability