on the well posedness for nls in hs abstract in this article we prove the local well posedness of the cauchy problem for the nonlinear schrodinger equation i t u u c u u in h s r n with 0 4 n 2 s and 1 s min n 2 2

strichartz type estimates in mixed besov spaces with application to critical nonlinear schrodinger equations abstract we study the cauchy problem for the nonlinear schrodinger equation with power nonlinearity in the fractional order sobolev space h s r n we consider the case where the nonlinear term is h s critical and the differentiability thereof is less than s we prove the existence of time global solutions of the cauchy problem for small initial data to this end we prove strichartz type estimates for linear inhomogeneous schrodinger equations in mixed besov spaces

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

critical exponent for evolution equations in modulation spaces abstract in this paper we propose a method to find the critical exponent for certain evolution equations in modulation spaces we define an index s q and use it to determine the critical exponent of the fractional heat equation as an example we prove that when s q is greater than the critical exponent this equation is locally well posed in the space c 0 t m p q s and when s q is less than the critical exponent this equation is ill posed in the space c 0 t m 2 q s our method may further be applied to some other evolution equations

lattices and k3 surfaces of degree 6 we relate a class of degenerations of k3 surfaces of degree 6 to certain lattices via hodge theory

global decaying solutions of a focusing energy critical heat equation in r4 abstract we study solutions of the focusing energy critical nonlinear heat equation u t u u 2 u in r 4 we show that solutions emanating from initial data with energy and h 1 norm below those of the stationary solution w are global and decay to zero via the concentration compactness plus rigidity strategy of kenig merle 33 34 first global such solutions are shown to dissipate to zero using a refinement of the small data theory and the l 2 dissipation relation finite time blow up is then ruled out using the backwards uniqueness of escauriaza seregin sverak 17 18 in an argument similar to that of kenig koch 32 for the navier stokes equations

the cauchy problem for heat equations with exponential nonlinearity abstract the cauchy problem for the semilinear heat equations is studied in the orlicz space exp l 2 r n where any power behavior of interaction works as a subcritical nonlinearity we prove the existence of global solutions for the semilinear heat equations with the exponential nonlinearity under the smallness condition on the initial data in exp l 2 r n

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

inviscid limits of the complex ginzburg landau equation in the inviscid limit the generalized complex ginzburg landau equation reduces to the nonlinear schrodinger equation this limit is proved rigorously with h1 data in the whole space for the cauchy problem and in the torus with periodic boundary conditions the results are valid for nonlinearities with an arbitrary growth exponent in the defocusing case and with a subcritical or critical growth exponent at the level of l2 in the focusing case in any spatial dimension furthermore optimal convergence rates are proved the proofs are based on estimates of the schrodinger energy functional and on gagliardo nirenberg inequalities

remark on the scattering operator for the cubic nonlinear dirac equation in three space dimensions abstract this paper is concerned with the scattering operator s for the three dimensional dirac equation with a cubic nonlinearity it follows from known results that s is well defined on a neighborhood of 0 in the sobolev space h r 3 c 4 for any 1 in the present paper we prove that for any m n and s max m there exists some neighborhood u of 0 in the weighted sobolev space h s m r 3 c 4 such that s u h s m r 3 c 4

analyticity of solutions to a quadratic system of schrodinger equations without mass resonance condition abstract we consider the cauchy problem for a quadratic system of schrodinger equations in space dimensions n 4 without mass resonance condition analytic smoothing effect for the quadratic system of schrodinger equations under the mass resonance condition for data which satisfy exponentially decaying condition has been studied in 14 in this study we construct global solutions in the analytic hardy space via the phase modulation operator based on the scale critical sobolev space h n 2 2 with data like the test functions of fourier hyperfunctions under the condition m 1 m 2 1 4 n m 1 between two masses m 1 and m 2

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

reflection groups in algebraic geometry after a brief exposition of the theory of discrete reflection groups in spherical euclidean and hyperbolic geometry as well as their analogs in complex spaces we present a survey of appearances of these groups in various areas of algebraic geometry

a convergence condition for the quadrilateral wilson element the paper deals with the convergence properties of the nonconforming quadrilateral wilson element which violates the patch test the convergence of the element is proved under a certain condition on mesh subdivisions without any modifications of the variational formulation this result extends the range of applicability of wilson s element the necessity of the proposed condition is also discussed

some remarks on strichartz estimates for homogeneous wave equation abstract we give several remarks on strichartz estimates for homogeneous wave equation with special attention to the cases of l x estimates radial solutions and initial data from the inhomogeneous sobolev spaces in particular we give the failure of the endpoint estimate l t 4 l x for n 2

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

modified scattering operator for the hartree fock equation abstract we study the scattering problem for the hartree fock equation hrf i t u 1 2 u f u t x r r n n 2 where u u 1 u n t is a c n n 2 valued unknown function and f u f 1 u f n u t denotes a nonlinear term whose j th element is defined by f j u r n v x y k 1 n u k y 2 u j x u j y u k y u k x d y where v x x 1 r is called a coulomb potential we show that if 1 2 then the modified scattering operator for the system hrf is well defined from a neighborhood at the origin in the space h 0 to a neighborhood at the origin in the space h 0 where h 0 k l 2 1 x 2 k 2 l 2

a schrodinger equation with time oscillating critical nonlinearity abstract in this paper we study the time oscillating critical nonlinear schrodinger equation i u t u t u 4 n 2 u 0 in r n n 3 where is a periodic function we show that for a given initial condition u 0 in h 1 the solution u converges as to the solution u of the limiting equation i u t u i u 4 n 2 u 0 with the same initial condition where i is the average of we also show that if the solution u is global and has a certain decay property then u is also global if is sufficiently large similar results for the subcritical problem are given by cazenave and scialom 2010 9

on the most algebraic k3 surfaces and the most extremal log enriques surfaces we shall characterize the unique k3 surface of discriminant 3 or 4 called the most algebraic k3 surfaces by vinberg in terms of the fixed locus of an automorphism on it based on this result we show that there is up to isomorphisms only one rational log eriques surface of type d 19 and one of type a 19

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

the global cauchy problem for the non linear klein gordon equation on demontre l unicite de solutions faibles et l existence et l unicite de solutions globales fortement continues dans l espace energie

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

bilinear estimates and applications to 2d nls the three bilinearities uv uv uv for functions u v r2 0 t 7 c are sharply estimated in function spaces xs b associated to the schrodinger operator i t these bilinear estimates imply local wellposedness results for schrodinger equations with quadratic nonlinearity improved bounds on the growth of spatial sobolev norms of finite energy global in time and blow up solutions of the cubic nonlinear schrodinger equation and certain generalizations are also obtained

a new error analysis for discontinuous finite element methods for linear elliptic problems the standard a priori error analysis of discontinuous galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm in this paper a new error analysis of discontinuous galerkin methods is developed using only the h k weak formulation of a boundary value problem of order 2k this is accomplished by replacing the galerkin orthogonality with estimates borrowed from a posteriori error analysis and by using a discrete energy norm that is well defined for functions in h k

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

analysis of some quadrilateral nonconforming elements for incompressible elasticity in this work some nonconforming elements on arbitrary quadrilateral meshes in solving incompressible elastic equations are analyzed a uniform optimal convergence rate is established at the incompressible limit nu 0 5 for both displacement and stresses or pressure in the case of incompressible flow

the alternating group of degree 6 in the geometry of the leech lattice and k3 surfaces the alternating group of degree 6 is located at the junction of three series of simple non commutative groups simple sporadic groups alternating groups and simple groups of lie type it plays a very special role in the theory of finite groups we shall study its new roles both in a finite geometry of a certain pentagon in the leech lattice and also in the complex algebraic geometry of k3 surfaces

some remarks on the nonlinear schrodinger equation with fractional dissipation we consider the cauchy problem for the l2 critical focussing nonlinear schrodinger equation with a fractional dissipation according to the order of the fractional dissipation we prove the global existence or the existence of finite time blowup dynamics with the log log blow up speed for u t l2

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

on the role of quadratic oscillations in nonlinear schrodinger equations ii the l 2 critical case we consider a nonlinear semi classical schrodinger equation for which quadratic oscillations lead to focusing at one point described by a nonlinear scattering operator the relevance of the nonlinearity was discussed by r carles c fermanian kammerer and i gallagher for l 2 supercritical power like nonlinearities and more general initial data the present results concern the l 2 critical case in space dimensions 1 and 2 we describe the set of non linearizable data which is larger due to the conformal invariance as an application we precise a result by f merle and l vega concerning finite time blow up for the critical schrodinger equation the proof relies on linear and nonlinear profile decompositions

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

approximation by quadrilateral finite elements we consider the approximation properties of finite element spaces on quadrilateral meshes the finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element it is known that for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l p and order r in w 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r we show by means of a counterexample that this latter condition is also necessary as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements

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

singular ring solutions of critical and supercritical nonlinear schrodinger equations abstract we present new singular solutions of the nonlinear schrodinger equation nls i t t r r r d 1 r r 2 0 1 d 2 d 2 these solutions collapse with a quasi self similar ring profile q i e q where q 1 l 1 t q r r m t l exp i 0 t d s l 2 s i l t 4 l r 2 1 r r m t 2 l t is the ring width that vanishes at the singularity r m t r 0 l t is the ring radius and 2 d 1 the blowup rate of these solutions is 1 1 for 2 d 2 and 1 d 0 1 and a square root with a loglog correction the loglog law when 2 and 1 d 0 therefore the nls has solutions that collapse with any blowup rate p for 1 2 p 1 this study extends the results of g fibich n gavish x wang new singular solutions of the nonlinear schrodinger equation physica d 211 2005 193 220 for 1 and d 2 and of p raphael existence and stability of a solution blowing up on a sphere for a l 2 super critical non linear schrodinger equation duke math j 134 2 2006 199 258 for 2 and d 2 to all 2 d 2 and 1 d

a new family of nonconforming finite elements on quadrilaterals abstract in this paper we generalize one first order nonconforming quadrilateral finite element proposed by lin tobiska and zhou to any odd order we construct degrees of freedom for shape function spaces for this family of elements and show their unisolvency in addition we present a medium a priori error analysis for this family of nonconforming elements on general quadrilateral meshes compared with the classical error analysis of the nonconforming finite element method the a priori analysis herein only needs the h 1 regularity of the exact solution numerics are presented to demonstrate theoretical results which in particular show dependence of convergence on mesh distortion parameters

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

limit of the solution of a nonlinear schrodinger equation at blow up time we consider the solution of the equation i v t and 0 where v is defined on 0 t rn and is a nonlinear complex valued function let us denote the function x v t x by v t we suppose that v blows up in the norm h1 at time t under suitable assumptions on and we prove the existence of a limit for v t when t t in appropriate spaces furthermore we specify the behavior of v near blow up time

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

the period map for cubic fourfolds the period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20 we determine the image of this period map thus confirming a conjecture of hassett and give at the same time a new proof of the theorem of voisin that asserts that this period map is an open embedding an algebraic version of our main result is an identification of the algebra of sl 6 invariant polynomials on the representation space sym 3 6 with a certain algebra of meromorphic automorphic forms on a symmetric domain of orthogonal type of dimension 20 we also describe the stratification of the moduli space of semistable cubic fourfolds in terms of a vinberg dynkin diagram

local well posedness for the h 2 critical nonlinear schrodinger equation in this paper we consider the nonlinear schrodinger equation iu t delta u lambda u frac 4 n 4 u in r n n ge 5 with lambda in c we prove local well posedness local existence unconditional uniqueness continuous dependence in the critical space dot h 2 r n

p1 nonconforming quadrilateral finite element methods for second order elliptic problems a p1 nonconforming quadrilateral finite element is introduced for second order elliptic problems in two dimensions unlike the usual quadrilateral nonconforming finite elements which contain quadratic polynomials or polynomials of degree greater than 2 our element consists of only piecewise linear polynomials that are continuous at the midpoints of edges one of the benefits of using our element is convenience in using rectangular or quadrilateral meshes with the least degrees of freedom among the nonconforming quadrilateral elements an optimal rate of convergence is obtained also a nonparametric reference scheme is introduced in order to systematically compute stiffness and mass matrices on each quadrilateral an extension of the p1 nonconforming element to three dimensions is also given finally several numerical results are reported to confirm the effective nature of the proposed new element

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

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

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

continuous dependence of cauchy problem for nonlinear schrodinger equation in hs abstract we consider the cauchy problem for the nonlinear schrodinger equation i t u u 0 u 1 u u in r n where 0 1 c in h s subcritical and critical case 0 4 n 2 s when 1 s n 2 and 0 when s n 2 we show that the solution depends continuously on the initial value in the standard sense in h s r n if satisfies certain assumptions

hs global well posedness for semilinear wave equations abstract we consider the cauchy problem for semilinear wave equations in r n with n 3 making use of bourgain s method in conjunction with the endpoint strichartz estimates of keel and tao we establish the hs global well posedness with s