on hartree equations with derivatives abstract we consider the cauchy problem of two types of hartree equations with exchange correlation correction terms i u t u v k u u in r 1 n k 1 2 u 0 in r n n 1 where v 1 u x 1 u 2 2 u 2 v 2 u x u 2 we establish the well posedness of cauchy problems and show the smoothing effect of solutions for each 0 n and 0 1

a local smoothing estimate for the schrodinger equation we show that the schrodinger operator eit is bounded from to for all a 2n 1 2 1 q 2 q and q2 4 n 1 this is almost sharp with respect to the sobolev index we also show that the schrodinger maximal operator sup0 s0 if and only if it is bounded from to when s 2s0 a corollary is that sup0 3 4

an endpoint space time estimate for the schrodinger equation abstract we obtain endpoint estimates for the schrodinger operator f e i t f in l x q r n l t r r with initial data f in the homogeneous sobolev space h s r n the exponents and regularity index satisfy n 1 q 1 r n 2 and s n 2 n q 2 r for n 2 we prove the estimates in the range q 16 5 and for n 3 in the range q 2 4 n 1

smoothing estimates for the schrodinger equation with unbounded potentials abstract we prove a local in time smoothing estimate for a magnetic schrodinger equation with coefficients growing polynomially at spatial infinity the assumptions on the magnetic field are gauge invariant and involve only the first two derivatives the proof is based on the multiplier method and no pseudodifferential techniques are required

transference of local to global l2 maximal estimates for dispersive partial differential equations abstract in this paper we give an elementary proof for transference of local to global maximal estimates for dispersive pdes this is done by transferring local l 2 estimates for certain oscillatory integrals with rough phase functions to the corresponding global estimates the elementary feature of our approach is that it entirely avoids the use of the wave packet techniques which are quite common in this context and instead is based on scalings and classical oscillatory integral estimates

a strong type 2 2 estimate for a maximal operator associated to the schrodinger equation soit t f x sup t 0 t t f x ou t t f 1 exp it 2 f 1 4 on montre que etant donne un intervalle fini i i t f 2 dx c i r f x 2 dx

on the defect of compactness for the strichartz estimates of the schrodinger equations abstract in this paper we prove that every sequence of solutions to the linear schrodinger equation with bounded data in h1 r d d 3 can be written up to a subsequence as an almost orthogonal sum of sequences of the type h d 2 2nv t tn h2n x xn hn where v is a solution of the linear schrodinger equation with a small remainder term in strichartz norms using this decomposition we prove a similar one for the defocusing h1 critical nonlinear schrodinger equation assuming that the initial data belong to a ball in the energy space where the equation is solvable this implies in particular the existence of an a priori estimate of the strichartz norms in terms of the energy

a smoothing property of schrodinger equations in the critical case this paper deals with the critical case of the global smoothing estimates for the schrodinger equation although such estimates fail for critical orders of weights and smoothing it is shown that they are still valid if one works with operators with symbols vanishing on a certain set the geometric meaning of this set is clarified in terms of the hamiltonian flow of the laplacian the corresponding critical case of the limiting absorption principle for the resolvent is also established obtained results are extended to dispersive equations of schrodinger type to hyperbolic equations and to equations of other orders

propagation of singularities and maximal functions in the plane in this work we mainly generalize bourgain s circular maximal function to include variable coefficient averages our techniques involve a combination of bourgain s basic ideas plus microlocal analysis in particular to see the role of curvature we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations we also show that forp 2 there is local smoothing inl p for solutions to the wave equation

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

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

on some schrodinger and wave equations with time dependent potentials the existence and uniqueness of solutions in the initial value problem for schrodinger and wave equations in the presence of a large time dependent potential is studied the usual strichartz estimates for such linear evolutions are shown to hold true with optimal assumptions on the potentials as a byproduct one obtains a counterexample to the two dimensional double endpoint inhomogeneous strichartz estimate

sharp strichartz estimates on nontrapping asymptotically conic manifolds we obtain the strichartz inequalities u l q t l r x 0 1 m c u 0 l 2 m for any smooth n dimensional riemannian manifold m which is asymptotically conic at infinity with either short range or long range metric perturbation and nontrapping where u is a solution to the schrodinger equation iu t 1 2 m u 0 and 2 q r are admissible strichartz exponents 2 q n r n 2 this corresponds with the estimates available for euclidean space except for the endpoint q r 2 2 n n 2 when n 2 these estimates imply existence theorems for semi linear schrodinger equations on m by adapting arguments from cazenave and weissler and kato this result improves on our previous result which was an l 4 t x strichartz estimate in three dimensions it is closely related to results of staffilani tataru burq robbiano zuily and tataru who consider the case of asymptotically flat manifolds

uniqueness properties of solutions of schrodinger equations under suitable assumptions on the potentials v and a we prove that if u c 0 1 h1 is a solution of the linear schrodinger equation i t x u vu a xu on rd 0 1 and if u 0 in x r 0 1 for some r 0 then u 0 in rd 0 1 as a consequence we obtain uniqueness properties of solutions of nonlinear schrodinger equations of the form i t x u g x t u u xu xu on rd 0 1 where g is a suitable nonlinear term the main ingredient in our proof is a carleman inequality of the form e x1 v lx2lt2 e x1 xv bx 2lt2 c e x1 i t x v bx1 2lt2 for any v c r h1 with v t 0 for t 0 1 in this inequality bx 2 and bx1 2 are banach spaces of functions on rd and e x1 is a suitable weight

convergence of sequences of schrodinger means abstract we study convergence almost everywhere of sequences of schrodinger means we also replace sequences by uncountable sets

the general quasilinear ultrahyperbolic schrodinger equation abstract we consider the general quasilinear schrodinger equation whose second order coefficients are given by a real symmetric non degenerate matrix we deduce conditions which guarantee that the associated initial value problem is locally well posed

price s law on nonstationary space times abstract in this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t 3 local uniform decay rate price s law price 1972 54 for linear waves as a corollary we also prove price s law for certain small perturbations of the kerr metric this result was previously established by the second author in tataru 65 on stationary backgrounds the present work was motivated by the problem of nonlinear stability of the kerr schwarzschild solutions for the vacuum einstein equations which seems to require a more robust approach to proving linear decay estimates

analytic smoothing effect for the schrodinger equation with long range perturbation we study microlocal analytic singularity of solutions to schr odinger equation with analytic coefficients using microlocal weight estimate developped for estimating the phase space tunneling we prove microlocal smoothing estimates that generalize results by l robbiano and c zuily we suppose the schr odinger operator is a long range type perturbation of the laplacian and we employ positive commutator type estimates to prove the smoothing property

well posedness and smoothing effect of schrodinger poisson equation in this work we take under consideration the cauchy problem for the schrodinger poisson type equation i tu x2u v u u f u 2 u where f represents a local nonlinear interaction we take into account both attractive and repulsive models and v is taken as a suitable solution of the poisson equation v 1 2 x c u 2 c cc is the doping profile or impurities we show that this problem is locally well posed in the weighted sobolev spaces hs hs r 1 x2 1 2 2 with s 1 which means the local existence uniqueness and continuity of the solution with respect to the initial data moreover under suitable assumptions on the local interaction we show the existence of global solutions finally we establish that for s 1 local in time and space smoothing effects are present in the solution more precisely in this problem there is locally a gain of half a derivative

smoothing effects and dispersion of singularities for the schrodinger evolution group we describe smoothing effects and dispersion of singularities for the schrodinger evolution group in the weighted sobolev spaces under a fairly general assumption on the potential it is shown that all singularities in the wavefunction vanish instantly whenever the initial state has sufficient decay we measure the regularity gained by the wavefunction by the decay property of the initial state no assumptions on the regularity of the initial state are imposed throughout the paper

local smoothing for scattering manifolds with hyperbolic trapped sets we prove a resolvent estimate for the laplace beltrami operator on a scattering manifold with a hyperbolic trapped set and as a corollary deduce local smoothing we use a result of nonnenmacher zworski to provide an estimate near the trapped region a result of burq and cardoso vodev to provide an estimate near infinity and the microlocal calculus on scattering manifolds to combine the two

maximal function estimates of solutions to general dispersive partial differential equations let u x t sqf x t be the solution of the general dispersive initial value problem ot uiq d u o u x o f x x t ee rn x r and s f the global maximal operator of sqf sharp weighted lp esimates for sq f with f e h rtn are given for general phase functions q

global parametrices and dispersive estimates for variable coefficient wave equations in this article we consider variable coefficient time dependent wave equations in mathbb r times mathbb r n using phase space methods we construct outgoing parametrices and prove strichartz type estimates globally in time this is done in the context of c 2 metrics which satisfy a weak asymptotic flatness condition at infinity

smoothing estimates for non dispersive equations this paper describes an approach to global smoothing problems for non dispersive equations based on ideas of comparison principle and canonical transformation established in authors previous paper ruzhansky and sugimoto proc lond math soc 105 393 423 2012 where dispersive equations were treated for operators a d x of order m satisfying the dispersiveness condition nabla a xi ne 0 for xi not 0 the global smoothing estimate begin aligned left left langle x right rangle s d x m 1 2 e ita d x varphi x right l 2 left mathbb r t times mathbb r n x right le c left varphi right l 2 left mathbb r n x right quad s 1 2 end aligned is well known while it is also known to fail for non dispersive operators for the case when the dispersiveness breaks we suggest the estimate in the form begin aligned left left langle x right rangle s nabla a d x 1 2 e it a d x varphi x right l 2 left mathbb r t times mathbb r n x right le c left varphi right l 2 left mathbb r n x right quad s 1 2 end aligned which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator a d x we show that this estimate and its variants do continue to hold for a variety of non dispersive operators a d x where nabla a xi may become zero on some set moreover other types of such estimates and the case of time dependent equations are also discussed

smoothing effects for schrodinger equations with electro magnetic potentials and applications to the maxwell schrodinger equations abstract we consider schrodinger equations in r 1 2 with electro magnetic potentials the potentials belong to h 1 and typically they are time independent or determined as solutions to inhomogeneous wave equations we prove kato type smoothing estimates for solutions we also apply this result to the maxwell schrodinger equations in the lorentz gauge and prove unique solvability of this system in the energy space

pointwise convergence of solutions to schrodinger type equations abstract in this paper we study the pointwise convergence of the function e i t 2 f x whenever f belongs to some sobolev space when 2 e i t f x is the solution to the standard schrodinger equation and this problem has been widely studied here we focus on the case 0 1 the same convergence problem for the nonelliptic schrodinger equation is also discussed the proof here follows the idea of carleson s first paper on this subject but we do reprove some known theorems by this straightforward method and get some new results as well

nonlocal schrodinger equations in metric measure spaces abstract in this note we consider the pointwise convergence to the initial data for the solutions of some nonlocal dyadic schrodinger equations on spaces of homogeneous type we prove the a e convergence when the initial data belongs to a dyadic version of an l 2 based besov space

dimension of divergence sets for the pointwise convergence of the schrodinger equation abstract in this note we consider the pointwise convergence along curves for the schrodinger equation and obtain estimates for the capacitary dimension of divergence sets which extend our previous result in 6

resolvent estimates and local decay of waves on conic manifolds we consider manifolds with conic singularites that are isometric to r outside a compact set under natural geometric assumptions on the cone points we prove the existence of a logarithmic resonancefree region for the cut off resolvent the estimate also applies to the exterior domains of non trapping polygons via a doubling process the proof of the resolvent estimate relies on the propagation of singularities theorems of melrose and the second author 23 to establish a very weak huygens principle which may be of independent interest as applications of the estimate we obtain a exponential local energy decay and a resonance wave expansion in odd dimensions as well as a lossless local smoothing estimate for the schrodinger equation

endpoint strichartz estimates for the magnetic schrodinger equation abstract we prove strichartz estimates for the schrodinger equation with an electromagnetic potential in dimension n 3 the decay and regularity assumptions on the potentials are almost critical i e close to the coulomb case in addition we require repulsivity and a nontrapping condition which are expressed as smallness of suitable components of the potentials while the potentials themselves can be large the proof is based on smoothing estimates and new sobolev embeddings for spaces associated to magnetic potentials

strichartz estimates and local smoothing estimates for asymptotically flat schrodinger equations abstract in this article we study global in time strichartz estimates for the schrodinger evolution corresponding to long range perturbations of the euclidean laplacian this is a natural continuation of a recent article d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 of the third author where it is proved that local smoothing estimates imply strichartz estimates by d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 the local smoothing estimates are known to hold for small perturbations of the laplacian here we consider the case of large perturbations in three increasingly favorable scenarios i without non trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term ii with non trapping assumptions we prove global estimates modulo a lower order spatially localized error term and iii for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum

estimates for multiparameter maximal operators of schrodinger type abstract multiparameter maximal estimates are considered for operators of schrodinger type sharp and almost sharp results that extend work by rogers and villarroya are obtained we provide new estimates via the integrability of the kernel which naturally appears with a t t argument and discuss the behavior at the endpoints we treat in particular the case of global integrability of the maximal operator on finite time for solutions to the linear schrodinger equation and make some comments on an open problem

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

globally well and ill posedness for non elliptic derivative schrodinger equations with small rough data abstract we show that there exists s c 0 such that the cubic quartic non elliptic derivative schrodinger equations with small data in modulation spaces m 2 1 s r n for n 3 n 2 are globally well posed if s s c and ill posed if s s c in 2d cubic case using the gabor frame we get some time global dispersive estimates for the schrodinger semi group in anisotropic lebesgue spaces which include a time global maximal function estimate in the space l x 1 2 l x 2 t by resorting to the smooth effect estimate together with the dispersive estimates in anisotropic lebesgue spaces we show that the cubic hyperbolic derivative nls in 2d has a unique global solution if the initial data in feichtinger segal algebra or in weighted sobolev spaces are sufficiently small

space time estimates for the wave equations with singular potentials abstract we study the regularity of the solutions for the wave equations with potentials that are time dependent and singular the size of the potentials is exactly a function of the spatial dimension rather than being small enough in the known results based on a weighted l 2 estimate for the solutions we prove the local regularity and the strichartz estimates the solvability of the equation is also studied

local smoothing for the quantum liouville equation abstract we analyze in this work the regularity properties of the density operator solution to the quantum liouville equation as was recently done for the strichartz inequalities we extend to systems of orthonormal functions the local smoothing estimates satisfied by the solutions to the schrodinger equation we show in particular that the local density associated to the solution to the free linear quantum liouville equation admits locally fractional derivatives of given order provided the data belong to some schatten spaces

local smoothing property and strichartz inequality for schrodinger equations with potentials superquadratic at infinity abstract we study smoothing properties for time dependent schrodinger equations i u t 1 2 u v x u u 0 l 2 r n with potentials which satisfy v x o x m at infinity m 2 we show that the solution u t x is 1 m times differentiable with respect to x at almost all t r and explain that this is the result of the fact that the sojourn time of classical particles with energy in arbitrary compact set is less than ct 1 m during 0 t when is very large we also show strichartz s inequality with derivative loss for such potentials and give its application to nonlinear schrodinger equations

the cauchy problem for quasi linear schrodinger equations with x rn t r with x1 xn and summation convention one may think of the equation in 1 1 as a non linear schrodinger equation where the operator modeling the dispersion relation is non isotropic and depends also on the unknown function its conjugate and their gradients in the space variables under appropriate assumptions on the function f the method presented here extends to fully non linear schrodinger equations of the form

strichartz estimates via the schrodinger maximal operator we consider the schrodinger operator e it delta acting on initial data f in dot h s we show that an affirmative answer to a question of carleson concerning the sharp range of s for which lim t to 0 e it delta f x f x a e x in mathbb r n would imply an affirmative answer to a question of planchon concerning the sharp range of q and r for which e it delta is bounded in l x q mathbb r n l r t mathbb r when n 2 we unconditionally improve the range for which the mixed norm estimates hold

magnetic virial identities weak dispersion and strichartz inequalities we show a family of virial type identities for the schrodinger and wave equations with electromagnetic potentials as a consequence some weak dispersive inequalities in space dimension n 3 involving morawetz and smoothing estimates are proved finally we apply them to prove strichartz inequalities for the wave equation with a non trapping electromagnetic potential with almost coulomb decay

extremizability of fourier restriction to the paraboloid abstract in this article we prove that nearly all valid scale invariant fourier restriction inequalities for the paraboloid in r 1 d have extremizers and that l p normalized extremizing sequences are precompact modulo symmetries this result had previously been established for the case q 2 in the range where the boundedness of the restriction operator is still an open question our result is conditional on improvements toward the restriction conjecture

restriction and kakeya phenomena for finite fields the restriction and kakeya problems in euclidean space have received much attention in the last few decades and they are related to many problems in harmonic analysis partial differential equations pdes and number theory in this paper we initiate the study of these problems on finite fields in many cases the euclidean arguments carry over easily to the finite setting and are in fact somewhat cleaner but there are some new phenomena in the finite case which deserve closer study

dispersive smoothing effects for kdv type equations in this paper we study the smoothness properties of solutions of some nonlinear equations of korteweg de vries kdv type which are of the form tu a x t u3 f u2 u1 u x t 1 wherex r uj jxu andkandjare nonnegative integers our principal condition is thata x t be positive and bounded so that the dispersion is dominant it is shown under certain additional conditions onaandfthatc solutionsu x t are obtained fort 0 if the initial datau x 0 decays faster than it does polynomially onr and has certain initial sobolev regularity a quantitative relationship between the rate of decay and the amount of gain of smoothness is given lets0be the sobolev index if ru2 x 0 1 x m dx 2 for an integerm 0 and the solution obeys u hs0 for an existence time 0 t t thenu x t hmloc r for all 0 t t andu x t l1 0 t h m 1 loc r our method can also be extended to address the fully nonlinear dispersive equations related to 1

the schrodinger equation along curves and the quantum harmonic oscillator abstract we consider the schrodinger equation associated to the harmonic oscillator i t u h u where h x 2 with initial data in the sobolev space h s r d with d 2 and s 3 8 we prove almost everywhere convergence of the solution to its initial data as time tends to zero which improves a result of yajima 1990 30 to this end we consider almost everywhere convergence for the free schrodinger along curves as it turns out these problems are more or less equivalent to that of the free schodinger equation along vertical lines our results are obtained by showing the equivalence of related maximal estimates

smoothing properties for the higher order nonlinear schrodinger equation with constant coefficients abstract we study local and global existence and smoothing properties for the initial value problem associated to a higher order nonlinear schrodinger equation with constant coefficients which appears as a model for propagation of pulse in an optical fiber

smoothing properties of evolution equations via canonical transforms and comparison principle this paper describes a new approach to global smoothing problems for dispersive and non dispersive evolution equations based on the global canonical transforms and the underlying global microlocal analysis for this purpose the egorov type theorem is established with canonical transformations in the form of a class of fourier integral operators and their weighted l 2 boundedness properties are derived this allows us to globally reduce general dispersive equations to normal forms in one or two dimensions then a new comparison principle for evolution equations is introduced in particular it allows us to relate different smoothing estimates by comparing certain expressions involving their symbols as a result it is shown that the majority of smoothing estimates for different equations are equivalent to each other moreover new estimates as well as several refinements of known results are obtained the proofs are considerably simplified a comprehensive analysis is presented for smoothing estimates for dispersive equations applications are given to the detailed description of smoothing properties of the schrodinger relativistic schrodinger wave klein gordon and other equations

local well posedness for the modified kdv equation in almost critical widehat h r_s spaces we study the cauchy problem for the modified kdv equation u t u xxx u 3 x 0 u 0 u 0 for data u 0 in the space h r s defined by the norm u 0 h r s s u 0 lr local well posedness of this problem is established in the parameter range 2 r 1 s 1 2 1 2r so the case s r 0 1 which is critical in view of scaling considerations is almost reached to show this result we use an appropriate variant of the fourier restriction norm method as well as bi and trilinear estimates for solutions of the airy equation

effective limiting absorption principles and applications the limiting absorption principle asserts that if h is a suitable schrodinger operator and f lives in a suitable weighted l2 space namely h 0 1 2 sigma for some sigma 0 then the functions r lambda i varepsilon f h lambda i varepsilon 1 f converge in another weighted l2 space h 0 1 2 sigma to the unique solution u of the helmholtz equation h lambda u f which obeys the sommerfeld outgoing radiation condition in this paper we investigate more quantitative or effective versions of this principle for the schrodinger operator on asymptotically conic manifolds with short range potentials and in particular consider estimates of the form r lambda i varepsilon f h 0 1 2 sigma leq c lambda h f h 0 1 2 sigma we are particularly interested in the exact nature of the dependence of the constants c lambda h on both lambda and h it turns out that the answer to this question is quite subtle with distinctions being made between low energies lambda ll 1 medium energies lambda sim 1 and large energies lambda gg 1 and there is also a non trivial distinction between qualitative estimates on a single operator h possibly obeying some spectral condition such as non resonance or a geometric condition such as non trapping and quantitative estimates which hold uniformly for all operators h in a certain class using elementary methods integration by parts and ode techniques we give some sharp answers to these questions as applications of these estimates we present a global in time local smoothing estimate and pointwise decay estimates for the associated time dependent schrodinger equation as well as an integrated local energy decay estimate and pointwise decay estimates for solutions of the corresponding wave equation under some additional assumptions on the operator h

regularity of solutions to the schrodinger equation soit f dans l espace de schwartz s r n et soit s t f x u x t rn exp ix exp it 2 f d x r n t r f est la transformee de fourier de f on considere u solution de l equation de schrodinger u i u t on pose s f x sup 0 t 1 s t f x x r n on introduit des espaces de sobolev h s en posant h s f s f hd s s r ou f hd s rn 1 2 s f 2 d 1 2 on etudie pour quelles valeurs de s on a b s f x 2 dx 1 2 c b f hd s f s r n pour toutes les boucles b de r n

regularity and decay of solutions to the stark evolution equation long time behavior and regularity are studied for solutions of the stark equationut i x1 v x u u 0 x l2 rn it is shown that for a class of short range potentialsv x the gain of local smoothness and the decay as t are close to those of the corresponding schrodinger equationut i v x u

local smoothing and convergence properties of schrodinger type equations abstract the initial value problem for the schrodinger type equation u t ip d u u 0 x u 0 x x r n is considered here p is a principal type polynomial of order m i e p c 1 m 1 r or a certain type of other real symbol such as p m m 1 the following results are proved a if u 0 l 2 r n then for any fixed r 0 write u 0 u 1 u 2 where u 1 and u 2 are square integrable and the fourier transform of u 1 vanishes for r and that of u 2 vanishes for r if u 1 t x e itp d u 1 x u 2 t x e itp d u 2 x then u 1 is real analytic in r r n and satisfies for every multi index sup t x d u 1 t x c u 0 l 2 while 1 x 2 1 2 u 2 t x l 2 r t h m 1 2 r x n b if u 0 l 2 s r n s 0 then for every t 0 u t h loc m 1 s r n c if u 0 h s r n s 1 2 then for a e x r n u t x u 0 x as t 0

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

some dispersive estimates for schrodinger equations with repulsive potentials we prove the local smoothing effect for schrodinger equations with repulsive potentials for n 3 the estimates are global in time and are proved using a variation of morawetz multipliers as a consequence we give sharp constants to measure the attractive part of the potential and its rate of decay which turns out to be different whether dimension 3 or higher are considered also a notion of zero resonance arises in a natural way our smoothing estimate allows us to use sobolev inequalities and treat nonradial perturbations

cusum charts with variable sampling intervals a standard cumulative sum cusum chart for controlling the process mean takes samples from the process at fixed length sampling intervals and uses a control statistic based on a cumulative sum of differences between the sample means and the target value this article proposes a modification of the standard cusum scheme that varies the time intervals between samples depending on the value of the cusum control statistic the variable sampling interval vsi cusum chart uses short sampling intervals if there is an indication that the process mean may have shifted and long sampling intervals if there is no indication of a change in the mean if the cusum statistic actually enters the signal region then the vsi cusum chart signals in the same manner as the standard cusum chart a markov chain approach is used to evaluate properties such as the average time to signal and the average number of samples to signal results show that the proposed vsi cusum chart is considerably more efficient than the standard cus

on dyadic nonlocal schrodinger equations with besov initial data abstract in this paper we consider the pointwise convergence to the initial data for the schrodinger dirac equation i u t d u with u x 0 u 0 in a dyadic besov space here d denotes the fractional derivative of order associated to the dyadic distance on r the main tools are a summability formula for the kernel of d and pointwise estimates of the corresponding maximal operator in terms of the dyadic hardy littlewood function and the calderon sharp maximal operator

dispersive analysis of charge transfer models we prove dispersive estimates for the time dependent schrodinger equation with a charge transfer hamiltonian as a by product we also obtain another proof of asymptotic completeness of the wave operators for a charge transfer model established earlier by k yajima and j m graf we also consider a more general matrix non self adjoint charge transfer problem this model appears naturally in the study of nonlinear multisoliton systems and is specifically motivated by the problem of asymptotic stability of multisoliton states of a nonlinear schrodinger equation 2004 wiley periodicals inc

on a class of solutions to the generalized derivative schrodinger equations ii abstract in this note we shall continue our study on the initial value problem associated for the generalized derivative schrodinger gdnls equation t u i x 2 u u x u x t r 0 1 and 1 inspiring by cazenave naumkin s works we shall establish the local well posedness for a class of data of arbitrary size in an appropriate weighted sobolev space thus removing the size restriction on the data required in our previous work the main new tool in the proof is the homogeneous and inhomogeneous versions of the kato smoothing effect for the linear schrodinger equation with lower order variable coefficients established by kenig ponce vega

microlocal dispersive smoothing for the schrodinger equation this paper establishes a connection between the microlocal smoothness of solutions of the initial value problem for schrodinger s equation and the global behavior of bicharacteristics of the principal symbol in particular localized initial data gives rise to solutions which are microlocally smooth at all points which are not trapped backwards by the bicharacteristic flow the origin of the phenomenon is in the dispersive nature of the equation the results imply microlocal regularity properties of the fundamental solution 1995 john wiley sons inc

best constants in some operator smoothness estimates abstract we provide a short proof of the inequality cf ben artzi and klainerman regularity and decay of evolution equations preprint and kato and yajima rev math phys 1 1989 481 496 1 x 2 1 2 1 1 4 e it u 2 dt c u 2 with explicit essentially exact values for c

on the dimension of divergence sets of dispersive equations we refine results of carleson sjogren and sjolin regarding the pointwise convergence to the initial data of solutions to the schrodinger equation we bound the hausdorff dimension of the sets on which convergence fails for example with initial data in h 1 mathbb r 3 the sets of divergence have dimension at most one

an endline bilinear cone restriction estimate for mixed norms we prove an l 2 times l 2 rightarrow l t ql x p bilinear fourier extension estimate for the cone when p q are on the critical line 1 q frac n 1 2 1 1 p this extends previous results by wolff tao and lee vargas

taylor series expansion for solutions of the korteweg devries equation with respect to their initial values abstract the initial value problem ivp for the korteweg devries kdv equation t u u x u 3 x u 0 u x 0 x for x r t r establishes a nonlinear map k from the space hs r to the space c t t hs r it has been known for many years that this map k is continuous bona and smith 1975 kato 1983 and it was proved recently that this map is lipschitz continuous kenig et al 1993 it is shown in this paper that the nonlinear map k is infinitely many times frechet differentiable from hs r to c t t hs r furthermore it is proved that k has a taylor series expansion at any given hs r i e formula where kn the nth derivative of k at is an n linear map from the n fold space of hs r to c t t hs r and the series converges in the space c t t hs r uniformly for h for some 0 each term yn kn hn in the series solves a linearized kdv equation thus any solutions of the ivp can be obtained by solving a series of linear problems by contrast the corresponding map kp established by the initial value problem for the periodic kdv equation t u u x u 3 x u 0 u x 0 x for x s t r where s is the unit length circle in the plane is known to be continuous only from hs s to c t t hs s this is due to the lack of smoothing effects for solutions of the periodic kdv equation in this paper it is shown that kp is lipschitz continuous from hs 1 s to c t t hs s and is n times frechet differentiable from hs n 1 s to c t t hs s for any n 1 the method developed in this paper applies to other nonlinear dispersive wave equations

dispersion and moment lemmas revisited abstract we further investigate relations between dispersive effects like the morawetz inequality for various classical equations schrodinger dirac and wave equations after wigner transform these dispersive estimates are reduced to moment lemmas for kinetic equations they yield new results for the schrodinger valid up to the semiclassical limit wave and dirac equations radial pseudo differential operators and also kinetic equations

local smoothing effect on the schrodinger equation with harmonic potential abstract we consider the linear schrodinger equation with repulsive harmonic potential we establish the local smoothing effect of this type of equations our work extends the related results obtained by l vega and n visciglia for the free schrodinger equation

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

quasilinear schrodinger equations i small data and quadratic interactions abstract in this article we prove local well posedness in low regularity sobolev spaces for general quasilinear schrodinger equations these results represent improvements in the small data regime of the pioneering works by kenig ponce vega and kenig ponce rolvung vega where viscosity methods were used to prove existence of solutions in very high regularity spaces our arguments here are purely dispersive the function spaces in which we show existence are constructed in ways motivated by the results of mizohata ichinose doi and others including the authors

local smoothing for the schrodinger equation with a prescribed loss we consider a family of surfaces of revolution each with a single periodic geodesic which is degenerately unstable we prove a local smoothing estimate for solutions to the linear schrodinger equation with a loss that depends on the degeneracy and we construct explicit examples to show our estimate is saturated on a weak semiclassical time scale as a byproduct of our proof we obtain a cutoff resolvent estimate with a sharp polynomial loss

well posedness of cauchy problem for the fourth order nonlinear schrodinger equations in multi dimensional spaces abstract we study the well posedness of cauchy problem for the fourth order nonlinear schrodinger equations i t u e u 2 u p x u 2 x u 2 t r x r n where e 1 0 1 n 2 denotes the spatial dimension and p is a polynomial excluding constant and linear terms

coherence on fractals versus pointwise convergence for the schrodinger equation we consider carleson s problem regarding convergence for the schrodinger equation in dimensions d ge 2 we show that if the solution converges almost everywhere with respect to alpha hausdorff measure to its initial datum as time tends to zero for all data h s mathbb r d then s ge frac d 2 d 2 d 1 alpha this strengthens and generalises results of bourgain and dahlberg kenig

a new non interior continuation method for solving the second order cone complementarity problem abstract in this paper based on a symmetrically perturbed smoothing fischer burmeister function a non interior continuation method is proposed for solving the second order cone complementarity problem soccp the proposed algorithm solves only one linear system of equations and performs only one line search at each iteration under monotonicity it is shown that our algorithm is globally and locally superlinearly convergent without requiring strict complementarity assumption at the soccp solution furthermore the proposed algorithm has local quadratic convergence under mild conditions some numerical results are reported which indicate the effectiveness of the proposed algorithm