global well posedness and scattering for the defocusing l 2 critical nonlinear schrodinger equation when d 2 in this article we prove that the defocusing cubic nonlinear schrodinger initial value problem is globally well posed and scattering for u0 l2 r2 the proof uses the bilinear estimates of planchon and vega and a frequency localized interaction morawetz estimate similar to the high frequency estimate of colliander keel staffilani takaoka and tao and especially the low frequency estimate of dodson

global well posedness and scattering for the nonstandard defocusing beam equation abstract we consider the nonstandard defocusing beam equation u t t i 1 n i 4 u u u p 1 u 0 in the subcritical case we show the global well posedness and the scattering results in low dimensions by the compactness rigidity argument of kenig and merle 2006 2008 7 3 the new ingredient is that we establish the suitable strichartz estimate to overcome the difficulty caused by the complicated dispersive estimate

the focusing energy critical nonlinear schrodinger equation in dimensions five and higher we consider the focusing energy critical nonlinear schr odinger equation iu t delta u u 4 over d 2 u in dimensions d geq 5 we prove that if a maximal lifespan solution u colon i times bbb r d to bbb c obeys sup t in i nabla u t 2 nabla w 2 then it is global and scatters both forward and backward in time here w denotes the ground state which is a stationary solution of the equation in particular if a solution has both energy and kinetic energy less than those of the ground state w at some point in time then the solution is global and scatters we also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state similar results were obtained by kenig and merle for spherically symmetric initial data and dimensions d 3 4 5

global wellposedness of defocusing critical nonlinear schrodinger equation in the radial case is globally wellposed in time more precisely we obtain a unique solution u u ch1 0 such that for all time u t depends continuously on the data in fact the dependence is even real analytic here moreover there is scattering for t the same statement holds for radial data h s 1 and proves in particular global existence of classical solutions in the radially symmetric case also this issue was open thus this is the analogue for nls of the result for the wave equation with quintic nonlinearity obtained by m struwe str in the radial case and by m grillakis gr s s in general in the case of the wave equation the proof is based on the following two different facts

the stable topological hyperbolic space form problem for complete manifolds of finite volume about fifteen years ago a borel posed the following conjecture let m be a closed aspherical manifold i e rcim o for i 1 if g n m is a homotopy equivalence where n is another manifold then g is homotopic to a homeomorphism a stable version of this conjecture for m r a closed nonpositively curved manifold was verified in 4 corollary b namely if m is a closed non positively curved manifold and g n m is a homotopy equivalence where n is a manifold then

energy scattering for nonlinear klein gordon and schrodinger equations in spatial dimensions 1 and 2 we show that when n 1 and 2 the scattering operators are well defined in the whole energy space for nonlinear klein gordon and schrodinger equations in r1 n with nonlinearity u p 1 u p 1 4 n such results have been known only for n 3

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

semilinear schrodinger flows on hyperbolic spaces scattering in h 1 we prove global well posedness and scattering in h1 for the defocusing nonlinear schrodinger equations left begin array ll i partial t delta g u u u 2 sigma u 0 phi end array right on the hyperbolic spaces mathbb h d d 2 for exponents sigma in 0 2 d 2 the main unexpected conclusion is scattering to linear solutions in the case of small exponents for comparison on euclidean spaces scattering in h1 is not known for any exponent sigma in 1 d 2 d and is known to fail for sigma in 0 1 d our main ingredients are certain noneuclidean global in time strichartz estimates and noneuclidean morawetz inequalities

global well posedness and scattering for the energy critical defocusing hartree equation for radial data abstract we consider the defocusing h 1 critical hartree equation for the radial data in all dimensions n 5 we show the global well posedness and scattering results in the energy space the new ingredient in this paper is that we first take advantage of the term i x a i 1 2 u 2 1 x d x d t in the localized morawetz identity to rule out the possibility of energy concentration instead of the classical morawetz estimate dependent of the nonlinearity

blow up of rough solutions to the fourth order nonlinear schrodinger equation abstract this paper deals with the formation of singularities of rough blow up solutions to the fourth order nonlinear schrodinger equation the limiting profile and l 2 concentration of the rough blow up solutions are obtained in h s r 4 with s s 0 where s 0 9 721 20 1 793 the new ingredient relies on the refined compactness result developed by zhu et al s h zhu j zhang h yang limiting profile of the blow up solutions for the fourth order nonlinear schrodinger equation dyn partial differ equ 7 2010 187 205

dynamic algorithm for parameter estimation and its applications we consider a dynamic method based on synchronization and adaptive control to estimate unknown parameters of a nonlinear dynamical system from a given scalar chaotic time series we present an important extension of the method when the time series of a scalar function of the variables of the underlying dynamical system is given we find that it is possible to obtain synchronization as well as parameter estimation using such a time series we then consider a general quadratic flow in three dimensions and discuss the applicability of our method of parameter estimation in this case in practical situations one expects only a finite time series of a system variable to be known we show that the finite time series can be repeatedly used to estimate unknown parameters with an accuracy that improves and then saturates to a constant value with repeated use of the time series finally we suggest an important application of the parameter estimation method we propose that the method can be used to confirm the correctness of a trial function modeling an external unknown perturbation to a known system we show that our method produces exact synchronization with the given time series only when the trial function has a form identical to that of the perturbation

decay estimates and strichartz estimates of fourth order schrodinger operator abstract we study time decay estimates of the fourth order schrodinger operator h 2 v x in r d for d 3 and d 5 we analyze the low energy and high energy behaviour of resolvent r h z and then derive the jensen kato dispersion decay estimate and local decay estimate for e i t h p a c under suitable spectrum assumptions of h based on jensen kato type decay estimate and local decay estimate we obtain the l 1 l estimate of e i t h p a c in 3 dimension by ginibre argument and also establish the endpoint global strichartz estimates of e i t h p a c for d 5 furthermore using the local decay estimate and the georgescu larenas soffer conjugate operator method we prove the jensen kato type decay estimates for some functions of h

topological nonrigidity of nonuniform lattices we say that an arbitrary manifold m m is topologically rigid relative to its ends if it satisfies the following condition if n n is any other manifold with a compact subset c n for which a proper homotopy equivalence h n n m m is a homeomorphism on n n c then there is a compact subset k n and a proper homotopy ht n n m m from h to a homeomorphism such that ht and h agree on n n k for all t 0 1 we say that a manifold m without boundary is properly rigid or absolutely topologically rigid if we eliminate the requirements that h be a homeomorphism on n n c and agree with ht on n n k for all t 0 1 along the lines of the classical borel conjecture that all closed aspherical manifolds are topologically rigid farrell and jones 8 provide the following important theorem

quintic nls in the exterior of a strictly convex obstacle we consider the defocusing energy critical nonlinear schr odinger equation in the exterior of a smooth compact strictly convex obstacle in three dimensions for the initial value problem with dirichlet boundary condition we prove global well posedness and scattering for all initial data in the energy space

the dynamics of the 3d radial nls with the combined terms in this paper we show the scattering and blow up result of the radial solution with the energy below the threshold for the nonlinear schrodinger equation nls with the combined terms iu t delta u u 4 u u 2 u qquad qquad qquad qquad rm cnls in the energy space h 1 mathbb r 3 the threshold is given by the ground state w for the energy critical nls iu t u u 4 u this problem was proposed by tao visan and zhang in comm pdes 32 1281 1343 2007 the main difficulty is lack of the scaling invariance illuminated by ibrahim et al in analysis pde 4 3 405 460 2011 we need to give the new radial profile decomposition with the scaling parameter then apply it to the scattering theory our result shows that the defocusing dot h 1 subcritical perturbation u 2 u does not affect the determination of the threshold of the scattering solution of cnls in the energy space

global existence and asymptotic behavior of the cauchy problem for fourth order schrodinger equations with combined power type nonlinearities abstract we study the global existence of solutions in sobolev spaces and the asymptotic behavior for the cauchy problem of the following fourth order schrodinger equation with combined power type nonlinearities i u t 2 u 1 u p u 2 u 8 n 4 u 0 where t r x r n 8 n p 8 n 4 1 2 are nonzero real numbers

global existence and scattering for rough solutions of a nonlinear schr odinger equation on r 3 we prove global existence and scattering for the defocusing cubic nonlinear schrodinger equation in h s r 3 for s 4 the main new estimate in the argument is a morawetz type inequality for the solution this estimate bounds k x t k l 4 r3 r whereas the well known morawetz type estimate of lin strauss controls r 1 0 r r3 x t 4 x dxdt

stochastic nonlinear schrodinger equations abstract this paper is devoted to the well posedness of stochastic nonlinear schrodinger equations in the energy space h 1 r d and is a continuation of our recent work barbu et al 2014 we allow general complex coefficients in the noise thus covering both the conservative and non conservative case the latter being important to include quantum measurement effects in the underlying physical model we consider both focusing and defocusing nonlinearities and prove global well posedness in h 1 r d including also the pathwise continuous dependence on initial conditions with exponents of the nonlinearity in exactly the same sharp range as in the deterministic case in particular in the special conservative case this work improves earlier results in de bouard and debussche 2003 where the critical exponents could not be reached for d 6 in the defocusing and for d 4 in the focusing case respectively moreover the local existence uniqueness and blowup alternative are also established for the energy critical case the approach presented here is mainly based on the rescaling approach already used in barbu et al 2014 to study the l 2 case and also on the strichartz estimates established in marzuola et al 2008 for large perturbations of the laplacian

scattering of the energy critical nls with inverse square potential abstract we consider the initial value problem of the energy critical nonlinear schrodinger equation with an inverse square potential in dimension d 4 5 6 in the focusing case we achieve the global well posedness and scattering when the kinetic energy and energy of the solution are both below the threshold of the ground state solution in the defocusing case we prove that any h 1 bounded initial data leads to the global and scattering solution no radial assumption is made on the initial data

global well posedness and scattering for the defocusing energy critical nonlinear schrodinger equation in r1 4 we obtain global well posedness scattering uniform regularity and global l6t x spacetime bounds for energy space solutions to the defocusing energy critical nonlinear schrodinger equation in a 4 our arguments closely follow those of colliander keel et al though our derivation of the frequency localized interaction morawetz estimate is somewhat simpler as a consequence our method yields a better bound on the l6t x norm

almost sure local well posedness and scattering for the 4d cubic nonlinear schrodinger equation abstract we consider the cauchy problem for the defocusing cubic nonlinear schrodinger equation in four space dimensions and establish almost sure local well posedness and conditional almost sure scattering for random initial data in h x s r 4 with 1 3 s 1 the main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear schrodinger equation which is inspired by certain function spaces used in the study of the schrodinger maps problem and is based on strichartz spaces as well as variants of local smoothing inhomogeneous local smoothing and maximal function spaces additionally we prove an almost sure scattering result for randomized radially symmetric initial data in h x s r 4 with 1 2 s 1

the solutions for nonlinear schrodinger equations abstract we consider the nonlinear schrodinger equation i u t u f u 0 f u u 2 m u n m n n x t r n r u 0 x u 0 x we give a formal solution of the cauchy problem by this formal solution we obtain that there exists a constant b independent of p n such that for any initial data u 0 h s b 0 s n 2 2 m n 1 4 n 2 s the schrodinger equation has a unique global solution u l r b p 2 s moreover when m 0 we give a unique weak solution of the cauchy problem in d distribution

optimal bilinear control of stochastic nonlinear schrodinger equations mass sub critical case we study optimal control problems for stochastic nonlinear schrodinger equations in both the mass subcritical and critical case for general initial data of the minimal l 2 regularity we prove the existence and first order lagrange condition of an open loop control in particular these results apply to the stochastic nonlinear schrodinger equations with the critical quintic and cubic nonlinearities in dimensions one and two respectively furthermore we obtain uniform estimates of backward stochastic solutions in new spaces of type u 2 and v 2 adapted to evolution operators related to linear schrodinger equations with lower order perturbations these estimates yield a new temporal regularity of backward stochastic solutions which is crucial for the tightness of approximating controls induced by ekeland s variational principle

modular symbols and the topological nonrigidity of arithmetic manifolds in this paper we address the existence of smooth manifolds proper homotopy equivalent to nonuniform arithmetic manifolds m g k that are not homeomorphic to it while the manifolds m are properly rigid if rank 2 we show that the so called virtual structure group has infinite rank as a vector space if rank 4 2015 wiley periodicals inc

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

global well posedness and scattering for the defocusing mass critical nonlinear schrodinger equation for radial data in high dimensions we establish global well posedness and scattering for solutions to the defocusing mass critical pseudoconformal nonlinear schrodinger equation iut u u 4 nu for large spherically symmetric lx2 rn initial data in dimensions n 3 after using the concentration compactness reductions in 32 to reduce to eliminating blow up solutions that are almost periodic modulo scaling we obtain a frequency localized morawetz estimate and exclude a mass evacuation scenario somewhat analogously to 10 23 36 in order to conclude the argument