scattering versus blow up beyond the threshold for the focusing choquard equation abstract this note investigates the choquard equation i u u i u p u p 2 u 0 indeed above the mass energy threshold a scattering versus finite time blow up dichotomy is obtained in the mass super critical and energy sub critical regime

moment method analysis of finite rectangular waveguide phased arrays a method of moments formulation for small finite planar phased arrays of rectangular waveguides in an infinite ground plane is investigated piecewise sinusoidal uniform surface patches are used as the basis and testing functions for the unknown equivalent magnetic currents of each aperture the block toeplitz symmetry property for rectangular grid arrays is utilized in the solution of the system of equations data are presented for 11 times 11 arrays with various element sizes element spacings and scan angles good agreement with data in the literature is shown the analysis is particularly efficient for e plane and h plane scans for arrays having rectangular waveguide fed apertures excited by the dominant mode

on the cauchy problem for dispersive equations with nonlinear terms involving high derivatives and with arbitrarily large initial data the aim of this article is to study the cauchy problem for general nonlinear dispersive equations involving derivatives in the nonlinearity the use of some decay properties of the linear part allows us to address the case of arbitrarily large initial data

well posedness and existence of bound states for a coupled schrodinger gkdv system abstract we derive some new results concerning the cauchy problem and the existence of bound states for a class of coupled nonlinear schrodinger gkdv systems in particular we obtain the existence of strong global solutions for initial data in the energy space h 1 r h 1 r generalizing previous results obtained in tsutsumi 1993 11 corcho and linares 2007 13 and dias et al submitted for publication 14 for the nonlinear schrodinger kdv system

the radial defocusing energy supercritical nls in dimension four abstract we consider the radial defocusing nonlinear schrodinger equations i u t u u p u with supercritical exponent p 4 in four space dimensions and prove that any radial solution that remains bounded in the critical sobolev space must be global and scatter

on scattering for the defocusing high dimensional inter critical nls abstract in this paper we study the critical norm conjecture for the inter critical nonlinear schrodinger equation with critical index s c satisfying 1 2 s c 1 when d 5 under the assumption of uniform boundedness of the critical norm we prove the global well posedness and scattering for the cauchy problem we follow the standard concentration compactness rigidity method established in 15 16 and treat three scenarios for the critical element respectively moreover double duhamel method and interaction morawetz estimate are applied to exclude the critical element

global existence and blowup for a class of the focusing nonlinear schrodinger equation with inverse square potential abstract we consider a class of the focusing nonlinear schrodinger equation with inverse square potential i t u u c x 2 u u u u 0 u 0 h 1 t x r r d where d 3 4 d 4 d 2 and c 0 satisfies c d d 2 2 2 in the mass critical case 4 d we prove the global existence and blowup below ground states for the equation with d 3 and c d in the mass and energy intercritical case 4 d 4 d 2 we prove the global existence and blowup below the ground state threshold for the equation this extends similar results of 18 and 22 to any dimensions d 3 and a full range c d we finally prove the blowup below ground states for the equation in the energy critical case 4 d 2 with d 3 and c d 2 4 d d 2 2 d

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 existence for 2d nonlinear schrodinger equations via high low frequency decomposition method abstract we study global existence of solutions for the cauchy problem of the nonlinear schrodinger equation i u t u u 2 m u in the 2 dimension case where m is a positive integer m 2 using the high low frequency decomposition method we prove that if 10 m 6 10 m 5 s 1 then for any initial value h s r 2 the cauchy problem has a global solution in c r h s r 2 and it can be split into u t e i t y t with y c r h 1 r 2 satisfying y t h 1 1 t 2 1 s 10 m 5 s 10 m 6 where is an arbitrary sufficiently small positive number

final state problem for a quadratic nonlinear schrodinger system in two space dimensions with mass resonance abstract we consider the asymptotic behavior of a solution to a quadratic nonlinear schrodinger system in two space dimensions we construct a particular solution which describes the charge dissipation phenomenon in spite of the mass preserving structure of the system this is a consequence of a new explicit solution to a system of ordinary differential equation which approximates asymptotic behavior of a solution to the system

energy critical nonlinear schrodinger equation with inverse square potential and subcritical perturbation abstract in this study we consider the global well posedness and scattering theory for the energy critical nonlinear schrodinger equation with inverse square potential and subcritical perturbation in three spatial dimensions we establish the global well posedness theory and scattering theory for a 1 4 1 25 and we improve the results obtained by zhang and zheng 19

construction of solutions to the subcritical gkdv equations with a given asymptotical behavior we consider the subcritical generalized korteweg de vries equation ut uxx u4 x 0 t x r let rj t x qcj x xj cjt j 1 n be n soliton solutions to this equation denote u t the kdv linear group and let v be in an adequate weighted sobolev space we construct a solution u t to the generalized korteweg de vries equation such that limt u t u t v j 1nrj t h1 0

fractional schrodinger equation solvability and connection with classical schrodinger equation abstract we consider the dirichlet boundary problem for semilinear fractional schrodinger equation with subcritical nonlinear term local and global in time solvability and regularity properties of solutions are discussed but our main task is to describe the connections of the fractional equation with the classical nonlinear schrodinger equation including convergence of the linear semigroups and continuity of the nonlinear semigroups when the fractional exponent approaches 1

martingale solutions for the stochastic nonlinear schrodinger equation in the energy space we consider a stochastic nonlinear schrodinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing stochastic nlse in h 1 on compact manifolds and bounded domains we construct a martingale solution using a modified faedo galerkin method based on the littlewood paley decomposition for the 2d manifolds with bounded geometry we use the strichartz estimates to show the pathwise uniqueness of solutions

solvability of the cauchy problem of non isotropic schrodinger equations in sobolev spaces abstract we study the existence of solutions for the cauchy problem of the non isotropically perturbed nonlinear schrodinger equation i u t u u u a u x 1 x 1 x 1 x 1 b u x 1 x 1 x 1 x 1 x 1 x 1 0 where a b are not simultaneously vanishing real constants is a positive constant and x x 1 x 2 r 2 by using kato s method we establish some local existence results for initial data belonging to h s r 2 where s 0 if either b 0 0 3 or b 0 a 0 0 8 3 s 1 3 if b 0 3 and s 1 3 8 if b 0 a 0 8 3 global existence is also established under some additional conditions

global existence of solutions for semi linear wave equation with scale invariant damping and mass in exponentially weighted spaces abstract in this paper we consider the following cauchy problem for the semi linear wave equation with scale invariant dissipation and mass and power non linearity u t t u 1 1 t u t 2 2 1 t 2 u u p u 0 x u 0 x u t 0 x u 1 x where 1 2 2 are nonnegative constants and p 1 on the one hand we will prove a global in time existence result for under suitable assumptions on the coefficients 1 2 2 of the damping and the mass term and on the exponent p assuming the smallness of data in exponentially weighted energy spaces on the other hand a blow up result for is proved for values of p below a certain threshold provided that the data satisfy some integral sign conditions combining these results we find the critical exponent for in all space dimensions under certain assumptions on 1 and 2 2 moreover since the global existence result is based on a contradiction argument it will be shown firstly a local in time existence result

the lagrangian averaged navier stokes equation with rough data in sobolev spaces abstract the lagrangian averaged navier stokes equation is a recently derived approximation to the navier stokes equation in this article we prove the existence of short time solutions to the incompressible isotropic lagrangian averaged navier stokes equation with low regularity initial data in sobolev spaces w s p r n for 1 p for l 2 based sobolev spaces we obtain global existence results more specifically we achieve local existence with initial data in the sobolev space h n 2 p p r n for initial data in h 3 4 2 r 3 we obtain global existence improving on previous global existence results which required data in h 3 2 r 3

a note on the fractional schrodinger equation of choquard type abstract some well posedness issues for a class of fractional schrodinger choquard equations are investigated first local well posedness is obtained in the energy space second the best constant of a gagliardo nirenberg type inequality is investigated third a sharp threshold of global existence versus blow up dichotomy is obtained finally the mass concentration is established for the blowing up solutions of the mass critical equation

nonlinear small data scattering for the generalized korteweg de vries equation abstract we study the longtime stability of small solutions to the ivp for the generalized korteweg de vries equation we obtain a lower bound for the degrees of nonlinearity of the perturbation which guarantees that the small solutions of the nonlinear problem behave asymptotically like the solutions of the associated linear problem this behavior allows us to establish a nonlinear scattering result for small perturbations with these degrees for a class of small data we improve the value of this lower bound the new crucial ingrediants in our proofs are the l p decay estimates of the half derivative of the airy kernel

on the well posedness of various one dimensional model equations for fluid motion abstract we consider 1d equations with nonlocal velocity of the form w t u w x u x w w where the nonlocal velocity u is given by 1 u 1 x x w 0 or 2 u h w h is the hilbert transform in this paper we address several local well posedness results with blow up criteria for smooth initial data we then establish the global well posedness by using the blow up criteria

existence and scattering of small solutions to a boussinesq type equation of sixth order abstract in this paper we consider the existence and uniqueness of the global small solution as well as the small data scattering result to the cauchy problem for a boussinesq type equation of sixth order with the nonlinear term f u behaving as u p p 1 as u 0 in r n n 1 the main method and techniques used in our paper are the littlewood paley dyadic decomposition the stationary phase estimate and some properties of bessel function

well posedness of cauchy problem of fourth order schrodinger equations in anisotropic sobolev space abstract we study the existence of solutions for the cauchy problem of the non isotropically perturbed nonlinear schrodinger equation iu t u a i 1 d u x i x i x i x i u u 0 where a is a real number a 0 is a positive constant 1 d n and x x 1 x 2 x n r n t r by using kato s method we establish local solvability results for initial data belonging to w 2 2 1 r n

further regularity results for almost cubic nonlinear schrodinger equation abstract we present three results related with the regularity of solutions of the almost cubic nls in the first one following ozawa s idea we establish mass and energy conservation for the solutions without regularizing the initial datum our second result is the h s well posedness for the cauchy problem for 0 s 1 finally we show that the same solutions are also in some bourgain spaces for possibly a smaller time interval in all of our results the non local nonlinear term in the equation is shown to act like a cubic nonlinearity on the appropriate sobolev and besov spaces

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

dispersion of small amplitude solutions of the generalized korteweg de vries equation abstract we study the long time behavior of small solutions of the initial value problem for the generalized korteweg de vries equation t u x 3 u x f u 0 gkdv u x 0 g x for the case where f w w s with s 1 4 23 57 3 8625 our results imply that if g l11 g l22 is sufficiently small then sup r 1 t 1 3 u t l in particular the solution tends to zero in the supremum norm the proofs make use of duhamel s formula and dispersion estimates for the linear propagator as well as chain and leibniz rules for fractional derivatives of compositions d f u lp and products d fg lp 0

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

on the interpolation with the potential bound for global solutions of the defocusing cubic wave equation on t2 abstract consider the solutions of the defocusing cubic wave equation t t u u u 3 with real data in h s t 2 h s 1 t 2 s 2 5 we prove that the solutions exist globally in time by contradiction assuming that one of the maximal times of existence is finite we prove that the sobolev norm of each of these solutions is bounded in an open neighborhood of it by using the i method see e g 4 it consists of introducing a mollified energy and estimating its growth in this neighborhood the growth is first estimated on small subintervals by means of dispersive bounds then it is estimated on the whole interval by an iteration process in order to enlarge the size of subintervals on which the dispersive bounds hold we interpolate with the potential bound for the low frequency part our results complement those obtained in 1

inviscid limit for the damped boussinesq equation abstract in this paper we consider the limit behavior of solutions to the cauchy problem for damped boussinesq equation in the regime of small viscosity in r n it is shown that the strong solutions of damped boussinesq equation converge to a strong solution of boussinesq equation in any time interval 0 t the key idea is that we only take into account the effect of dispersion and overlook the effect of dissipation in the damped boussinesq equation which is a kind of new view we analyze the dispersion by the techniques of high low frequency decomposition and littlewood paley dyadic decomposition stationary phase estimate and some properties of bessel function

remarks on modified improved boussinesq equations in one space dimension we study the existence and scattering of global small amplitude solutions to modified improved boussinesq equations in one dimension with nonlinear term graphic 1 behaving as a power graphic 2 as graphic 3 solutions are considered in graphic 4 space for all graphic 5 according to the value of s the power nonlinearity exponent p is determined liu liu 1996 indiana univ math j 45 797 816 obtained the minimum value of p greater than 8 at graphic 6 for sufficiently small cauchy data in this paper we prove that p can be reduced to be greater than graphic 7 at graphic 8 and the corresponding solution u has the time decay such as graphic 9 as graphic 10 we also prove non existence of non trivial asymptotically free solutions for graphic 11 under vanishing condition near zero frequency on asymptotic states 1 embed inline graphic 1 gif 2 embed inline graphic 2 gif 3 embed inline graphic 3 gif 4 embed inline graphic 4 gif 5 embed inline graphic 5 gif 6 embed inline graphic 6 gif 7 embed inline graphic 7 gif 8 embed inline graphic 8 gif 9 embed inline graphic 9 gif 10 embed inline graphic 10 gif 11 embed inline graphic 11 gif

an initial boundary value problem of 2d nonlinear schrodinger equation in a strip abstract the paper studies the initial boundary value problem ibvp of a nonlinear schrodinger equation posed on a strip domain r 0 1 with non homogeneous dirichlet boundary conditions for 0 s 1 2 if the initial condition x y is in sobolev space h s r 0 1 and the boundary condition h x t is in h s r 2 h x t l 2 r 2 1 1 2 1 2 s 2 h l 2 r 2 where h is the fourier transform of h with respect to t and x the local well posedness of the ibvp in c 0 t h s r 0 1 is proved for 1 2 s 5 2 because of compatibility conditions the same local well posedness result holds if an extra condition h x 0 h s 1 2 r is added moreover the global well posedness is obtained for s 1 the basic idea used here relies on the derivation of an integral operator for the non homogeneous boundary data and the proof of strichartz s estimates for this operator after the problem is transformed to finding a fixed point of an integral operator the contraction mapping argument then yields a fixed point using the strichartz s estimates for initial and boundary operators the global well posedness is proved using a priori estimates of the solutions

sobolev algebras on lie groups and riemannian manifolds we prove that on any connected unimodular lie group g the space l p g l g where l p g is the sobolev space of order 0 associated with a sublaplacian is an algebra under pointwise product this generalizes results due to strichartz in the euclidean case to bohnke in the case of stratified groups and others a global version of this fact holds for groups with polynomial growth we give similar results for riemannian manifolds with ricci curvature bounded from below respectively nonnegative

scattering above energy norm of solutions of a loglog energy supercritical schrodinger equation with radial data abstract we prove global well posedness and scattering of h k h k r n h 1 r n solutions of the loglog energy supercritical schrodinger equation i t u u u 4 n 2 u log c log 10 u 2 0 c c n n 3 4 with radial data u 0 u 0 h k h k r n h 1 r n k n 2 this is achieved roughly speaking by extending bourgain s argument in bourgain 1999 1 see also grillakis 2000 5 and tao s argument in tao 2005 10 in high dimensions

global well posedness and scattering for the defocusing energy critical nonlinear schrodinger equations of fourth order in dimensions d 9 abstract we consider the defocusing energy critical nonlinear schrodinger equation of fourth order i u t 2 u u 8 d 4 u we prove that any finite energy solution is global and scatters both forward and backward in time in dimensions d 9

rough blowup solutions to the l 2 critical nls we study the singularity formation for the cubic focusing l 2 critical nonlinear schrodinger equation on mathbb r 2 in a series of recent works merle and raphael have completely described the so called log log blowup regime and proven its stability in the energy space h 1 our aim in this paper is to investigate the stability of this blowup regime under rough perturbations in the direction of developing a theory at the level of the critical space l 2 by blending the merle raphael techniques with the quantitative i method developed by colliander keel staffilani takaoka and tao for the study of the cauchy problem for rough data we obtain the stability of the log log regime in h s for all s 0

the defocusing energy supercritical nls in four space dimensions abstract we consider a class of defocusing energy supercritical nonlinear schrodinger equations in four space dimensions following a concentration compactness approach we show that for 1 s c 3 2 any solution that remains bounded in the critical sobolev space h x s c r 4 must be global and scatter key ingredients in the proof include a long time strichartz estimate and a frequency localized interaction morawetz inequality

nonlinear schrodinger systems in 2d with nondecaying final data abstract we consider nonlinear schrodinger systems with quadratic nonlinearities in two space dimensions we prove the existence of modified wave operators and uniform time decay of solutions when the fourier transform of the final data does not necessarily decay at spatial infinity and is small in uniform norm

scattering in h1 for the intercritical nls with an inverse square potential abstract we study the nonlinear schrodinger equation with an inverse square potential in dimensions 3 d 6 we consider both focusing and defocusing nonlinearities in the mass supercritical and energy subcritical regime in the focusing case we prove a scattering blowup dichotomy below the ground state in the defocusing case we prove scattering in h 1 for arbitrary data

optimal convergence rate and regularity of nonrelativistic limit for the nonlinear pseudo relativistic equations abstract in this paper we are concerned with the nonrelativistic limit of the following pseudo relativistic equation with hartree nonlinearity or power type nonlinearity 2 c 2 m 2 c 4 m c 2 u u n u where c denotes the speed of light we prove that as c the ground states of this equation converges to the ground state of its nonrelativistic counterpart 2 2 m u u n u with convergence rate 1 c 2 in every h s norms moreover we show that this rate is optimal

strichartz estimates for the euler equations in the rotational framework abstract we consider the initial value problems of the incompressible euler equations in the rotational framework we obtain the optimal range of the strichartz estimate for the linear group associated with the coriolis forces as an application we prove that the lifespan of the solution can be taken arbitrarily large provided that the speed of rotation is sufficiently high

inward outward energy theory of non radial solutions to 3d semi linear wave equation abstract the topic of this paper is a semi linear energy sub critical defocusing wave equation t 2 u u u p 1 u in the 3 dimensional space with 3 p 5 we generalize inward outward energy theory and weighted morawetz estimates for radial solutions to the non radial case as an application we show that if 3 p 5 and 5 p 2 then the solution scatters as long as the initial data u 0 u 1 satisfy r 3 x 1 1 2 u 0 2 1 2 u 1 2 1 p 1 u 0 p 1 d x if p 3 we can also prove the scattering result if initial data u 0 u 1 are contained in the critical sobolev space and satisfy the inequality r 3 x 1 2 u 0 2 1 2 u 1 2 1 4 u 0 p 1 d x these assumptions on the decay rate of initial data as x are weaker than previously known scattering results in the non radial case

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

knapp stein type intertwining operators for symmetric pairs abstract for a symmetric pair g h of reductive groups we construct a family of intertwining operators between spherical principal series representations of g and h that are induced from parabolic subgroups satisfying certain compatibility conditions the operators are given explicitly in terms of their integral kernels and we prove convergence of the integrals for an open set of parameters and meromorphic continuation in particular this establishes lower bounds for multiplicities of intertwining operators we further discuss uniqueness of intertwining operators and for the rank one cases g h su 1 n f s u 1 m f u n m f with f r c h o and for the pair g h gl 4 n r gl 2 n c we show that for a certain choice of maximal parabolic subgroups our operators generically span the space of intertwiners

exact smoothing properties of schrodinger semigroups we study schrodinger semigroups in the scale of sobolev spaces and show that for kato class potentials the range of such semigroups in lp has exactly two more derivatives than the potential this proves a conjecture of b simon we show that eigenfunctions of schrodinger operators are generically smoother by exactly two derivatives in given sobolev spaces than their potentials we give applications to the relation between the potential s smoothness and particle kinetic energy in the context of quantum mechanics and characterize kinetic energies in coulomb systems the techniques of proof involve leibniz and chain rules for fractional derivatives which are of independent interest as well as a new characterization of the kato class 1 introduction in this paper we attempt a precise study of the action of schrodinger semigroups in the scale of sobolev spaces work in this area was begun by b simon si4 in 1985 he proved a number of positive and negative results on such smoothing and also presented an open question in the form of a conjecture regarding necessary and sufficient conditions on the potential v for smoothing of order s by the semigroup of functions in l2 rd study of schrodinger semigroups is of interest for a number of reasons see

scattering theory for the defocusing energy supercritical nonlinear wave equation with a convolution abstract in this paper we study the global well posedness and scattering problem for the nonlinear wave equation with a convolution u t t u x u 2 u 0 in dimensions d 6 we show that if the solution u is apriorily bounded in the critical homogeneous sobolev space that is u l t h x s c h x s c 1 with s c 2 2 1 then u is global and it scatters the impetus to consider this problem stems from a series of recent works for the energy supercritical nonlinear wave equation and nonlinear schrodinger equation our analysis derived from the concentration compactness method to show that the proof of the global well posedness and scattering is reduced to disprove the existence of three scenarios the finite time blow up solution the soliton like solution and the low to high frequency cascade we note that authors preclude the finite time blow up solution to wave equation usually by the property of the finite speed of propagation in the previous literature however the finite speed of propagation is broken in the nonlocal nonlinear wave equation with a convolution nonlinearity for this we will initially establish the low regularity results for almost periodic solutions including the finite time blow up solutions and initially preclude the finite time blow up solutions by lowering the regularity

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

well posedness of the cauchy problem of high dimension non isotropic fourth order schrodinger equations in sobolev spaces abstract we study the well posedness for the cauchy problem of the non isotropically perturbed nonlinear schrodinger equation i u t u u u a i 1 d u x i x i x i x i 0 where a is a real constant 1 d n is an integer is a positive constant and x x 1 x 2 x n r n by using kato s method we establish some local and global existence results for initial data belonging to h s r n where s 0 if 0 8 2 n d s n 2 1 8 2 n d if 8 2 n d

global well posedness and scattering for the defocusing energy supercritical cubic nonlinear wave equation abstract in this paper we consider the defocusing cubic nonlinear wave equation u t t u u 2 u 0 in the energy supercritical regime in dimensions d 6 with no radial assumption on the initial data we prove that if a solution satisfies an a priori bound in the critical homogeneous sobolev space throughout its maximal interval of existence that is u l t h x s c h x s c 1 then the solution is global and it scatters our analysis is based on the methods of the recent works of kenig and merle 2008 21 and killip and visan 2010 26 27 treating the energy supercritical nonlinear schrodinger and wave equations