long time behavior of solutions of gkdv equations abstract in this paper we consider the long time behavior of solutions of a version of gkdv equation on the real line the existence of the global attractor in h 2 r is proved when the force f belongs to h 2 r moreover we investigate the change of the attractor according to the change of f and the coefficient of damping term finally several numerical experiments confirm all the theoretical conclusions

hard lefschetz theorem for valuations complex integral geometry and unitarily invariant valuations we obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets a version of the hard lefschetz theorem using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations it implies new integral geometric formulas for real submanifolds in hermitian spaces generalizing the classical kinematic formulas in euclidean

eigenvalues of rank one perturbations of unstructured matrices abstract let a be a fixed complex matrix and let u v be two vectors the eigenvalues of matrices a uv r form a system of intersecting curves the dependence of the intersections on the vectors u v is studied

valuations on sobolev spaces all affinely covariant convex body valued valuations on the sobolev space w 1 1 bbb r n are completely classified it is shown that there is a unique such valuation for blaschke addition this valuation turns out to be the operator which associates with each function f in w 1 1 bbb r n the unit ball of its optimal sobolev norm

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

random attractors for the stochastic benjamin bona mahony equation on unbounded domains abstract we prove the existence of a compact random attractor for the stochastic benjamin bona mahony equation defined on an unbounded domain this random attractor is invariant and attracts every pulled back tempered random set under the forward flow the asymptotic compactness of the random dynamical system is established by a tail estimates method which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity

existence and regularity of the global attractor for a weakly damped forced shallow water equation in h1 r abstract in this paper we prove the existence of the global attractor for a weakly damped forced shallow water equation in h 1 r moreover we show that the global attractor is actually compact in h 4 r the main method is the combination of the energy equation a suitable splitting of the solutions and bilinear estimates in bourgain spaces

a characterization of affine surface area abstract we show that every upper semicontinuous and equi affine invariant valuation on the space of d dimensional convex bodies is a linear combination of affine surface area volume and the euler characteristic

pullback attractors for generalized korteweg de vries burgers equations abstract in this paper we study the long time behavior of solutions to the generalized korteweg de vries burgers equation of the form u t u x x x g u x u u x x f t where 0 0 we prove the existence of pullback attractors a in h 1 r for this equation in both cases 0 and 0 moreover the upper semicontinuity of the pullback attractors at 0 is also investigated

existence and upper semicontinuity of bi spatial pullback attractors for smoothing cocycles abstract in this work we establish several criteria for the existence as well as the upper semi continuity of bi spatial attractors under a closedness condition which dramatically weakens the usual requirement on the continuity of the cocycle it is also shown that though the continuity plays a less important role in the study of attractors it is impossible to establish an existence criteria for common attractors for systems without any continuity like properties however for such bad systems one can expect a mini attractor which is shown adequate well to depict the asymptotic behavior of non continuous systems finally we study the l 2 h 0 1 pullback attractor for a stochastic complex ginzburg landau equation a spectrum decomposition method is employed to overcome the lack of sobolev compactness embeddings in h 0 1

long time dynamics for a damped benjamin bona mahony equation in low regularity spaces abstract long time behavior of solutions for a damped bbm equation in low regularity space is investigated by the i method and high low frequency decomposition we prove the existence of a global attractor in l 2 if the force belongs to l 2 moreover we show that the global attractor has finite fractal dimension in the sharp regularity space h 2

ellipsoids and matrix valued valuations a classification is obtained of borel measurable gl n covariant symmetric matrix valued valuations on the space of n dimensional convex polytopes the only ones turn out to be the moment matrix corresponding to the classical legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by lutwak yang and zhang a classical concept from mechanics is the legendre ellipsoid or ellipsoid of inertia 2k associated with a convex body k r it can be defined as the unique ellipsoid centered at the center of mass of k such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of k if we fix a scalar product x y for x y r 2k can be defined by the moment matrix m2 k of k this is the n n matrix with coefficients

pullback attractors for the non autonomous fitzhugh nagumo system on unbounded domains abstract the existence of a pullback attractor is established for the singularly perturbed fitzhugh nagumo system defined on the entire space r n when external terms are unbounded in a phase space the pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far field values of solutions although the limiting system has no global attractor we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded and hence do not blow up as a small parameter approaches zero

the cramer rao inequality for star bodies associated with each body k in euclidean n space mathbb r sp n is an ellipsoid gamma sb 2k called the legendre ellipsoid of k it can be defined as the unique ellipsoid centered at the body s center of mass such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of the body in an earlier paper the authors showed that corresponding to each convex body k subset mathbb r sp n is a new ellipsoid gamma sb 2 k that is in some sense dual to the legendre ellipsoid the legendre ellipsoid is an object of the dual brunn minkowski theory while the new ellipsoid gamma sb 2 k is the corresponding object of the brunn minkowski theory the present paper has two aims the first is to show that the domain of gamma sb 2 can be extended to star shaped sets the second is to prove that the following relationship exists between the two ellipsoids if k is a star shaped set then gamma sb 2 k subset gamma sb 2k with equality if and only if k is an ellipsoid centered at the origin this inclusion is the geometric analogue of one of the basic inequalities of information theory the cramer rao inequality

global attractor for weakly damped nonlinear schrodinger equations in l2 r abstract we prove that the weakly damped nonlinear schrodinger flow in l 2 r provides a dynamical system which possesses a global attractor the proof relies on the continuity of the schrodinger flow for the weak topology in l 2 r

structure of the unitary valuation algebra s alesker has shown that if g is a compact subgroup of o n acting transitively on the unit sphere sn 1 then the vector space valg of continuous translation invariant g invariant convex valuations on rn has the structure of a finite dimensional graded algebra over r satisfying poincare duality we show that the kinematic formulas for g are determined by the product pairing using this result we then show that the algebra valu n is isomorphic to r s t fn 1 fn 2 where s t have degrees 2 and 1 respectively and the polynomial fi is the degree i term of the power series log 1 s t

eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations abstract udy the perturbation theory of structured matrices under structured rank one perturbations and then focus on several classes of complex matrices generic jordan structures of perturbed matrices are identified it is shown that the perturbation behavior of the jordan structures in the case of singular j hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one perturbation as it has been studied in 18 28 30 31 thus a generic structured perturbation would not be generic if considered as an unstructured perturbation in other settings of structured matrices the generic perturbation behavior of the jordan structures within the confines imposed by the structure follows the pattern of that of unstructured perturbations

asymptotic smoothing and attractors for the generalized benjamin bona mahony equation on r 3 abstract we study the asymptotic behavior of the solutions of the benjamin bona mahony equation defined on r 3 we first provide a sufficient condition to verify the asymptotic compactness of an evolution equation defined in an unbounded domain which involves the littlewood paley projection operators we then prove the existence of an attractor for the benjamin bona mahony equation in the phase space h 1 r 3 by showing the solutions are point dissipative and asymptotic compact finally we establish the regularity of the attractor and show that the attractor is bounded in h 2 r 3

generalized korteweg de vries equation in h1 r abstract we consider the generalized kdv equation u t u x x x g u x u f x 0 we prove its local and global solvability in the phase space h 1 r and next when 0 we study asymptotic behavior of solutions constructing a global attractor of the generalized kdv equation in h 1 r

fisher information and matrix valued valuations abstract all continuous and affinely contravariant matrix valued valuations on the sobolev space w 1 2 r n are completely classified it is shown that there is a unique such valuation this valuation turns out to be the fisher information matrix

existence of global attractors for the benjamin bona mahony equation in unbounded domains in this paper we investigate the asymptotic behavior of the benjamin bona mahony equation in unbounded domains we prove the existence of a global attractor when the equation is defined in a three dimensional channel the asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions

intersection bodies and valuations all gl n covariant star body valued valuations on convex polytopes are completely classified it is shown that there is a unique nontrivial such valuation this valuation turns out to be the so called intersection operator an operator that played a critical role in the solution of the busemann petty problem

a new ellipsoid associated with convex bodies it is shown that corresponding to each convex body there is an ellipsoid that is in a sense dual to the legendre ellipsoid of classical mechanics sharp affine isoperimetric inequalities are obtained between the volume of the convex body and that of its corresponding new ellipsoid these inequalities provide exact bounds for the isotropic constant associated with the new ellipsoid among other things this leads to a new approach to establishing ball s maximal shadows conjecture for symmetric convex bodies corresponding to each origin symmetric convex or more general subset of euclidean n space r there is a unique ellipsoid with the following property the moment of inertia of the ellipsoid and the moment of inertia of the convex set is the same about every 1 dimensional subspace of r this ellipsoid is called the legendre ellipsoid of the convex set the legendre ellipsoid and its polar the binet ellipsoid are well known concepts from classical mechanics see milman and pajor mpa1 mpa2 lindenstrauss and milman lim and leichtweis le for some historical references it has slowly come to be recognized that along side the brunn minkowski theory there is a dual theory the nature of the duality between this dual brunn minkowski theory and the brunn minkowski theory is subtle and not yet understood it is easily seen that the legendre and binet ellipsoid is an object of this dual brunnminkowski theory this observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical legendre ellipsoid in the brunn minkowski theory it is the aim of this paper to demonstrate the existence of precisely this dual object in retrospect one may well wonder why research supported in part by nsf grant dms 9803261 typeset by ams tex 1 2 a new ellipsoid associated with convex bodies the new ellipsoid presented in this note was not discovered long ago the simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of l2 curvature in hand however the brunn minkowski theory was only recently extended to incorporate the new notion of lp curvature see l2 l3 a positive definite n n real symmetric matrix a generates an ellipsoid a in r defined by a x r x ax 1 where x ax denotes the standard inner product of x and ax in r associated with a star shaped about the origin set k r is its legedre ellipsoid 2k which is generated by the matrix mij k where mij k n 2 v k k ei x ej x dx with e1 en denoting the standard basis for r and v k denoting the ndimensional volume of k we will associate a new ellipsoid 2k with each convex body k r one approach to defining 2k without introducing new notation is to first define it for polytopes and then use approximation with respect to the hausdorff metric to extend the definition to all convex bodies suppose p r is a polytope that contains the origin in its interior let u1 un denote the outer unit normals to the faces of p let a1 an denote the areas i e n 1 dimensional volumes of the corresponding faces and let h1 hn denote the distances from the origin to the corresponding faces the ellipsoid 2p is generated by the matrix mij p where mij p 1 v p n l 1 al hl ei ul ej ul an alternate definition of the operator 2 will be given after additional notation is introduced the easily established affine nature of the operator 2 is formally stated in lemma 1 if k r is star shaped about the origin then for each gl n

the global attractor of the viscous fornberg whitham equation abstract this paper aims to present a proof of the existence of the attractor for the one dimensional viscous fornberg whitham equation in this paper the global existence of solution to the viscous fornberg whitham equation in l 2 under the periodic boundary conditions is studied by using the time estimate of the fornberg whitham equation we get the compact and bounded absorbing set and the existence of the global attractor for the viscous fornberg whitham equation

valuations with crofton formula and finsler geometry valuations admitting a smooth crofton formula are studied using geometric measure theory and rumin s cohomology of contact manifolds the main technical result is a current representation of a valuation with a smooth crofton formula a geometric interpretation of alesker s product is given for such valuations as a first application in finsler geometry a short proof of the theorem of gelfand smirnov that crofton densities are projective is derived the holmes thompson volumes in a projective finsler space are studied it is shown that they induce in a natural way valuations and that the alesker product of the k dimensional and the l dimensional holmes thompson valuation is the k l dimensional holmes thompson valuation

higher regularity of global attractors of a weakly dissipative fractional korteweg de vries equation in this paper we study the long time behavior of solutions to a weakly dissipative fractional korteweg de vries kdv equation on the real line r the main difficulty lies in that the dissipative term is a nonlocal operator we overcome this difficulty by the commutator estimates and product estimates associated with fractional laplacian the asymptotical compactness of solution semigroup is proved by the tail estimates finally we conclude the existence of h2 r h5 r global attractor of the weakly dissipative fractional kdv equation

upper semicontinuity of attractors for nonclassical diffusion equations in h1 r3 abstract in this paper we mainly investigate upper semicontinuity of global attractors in h 1 r 3 for a perturbed nonclassical diffusion equations for this purpose some necessary abstract results are established

on the number of solutions to the discrete two dimensional l0 minkowski problem abstract our main result shows the uniqueness of planar convex polygons of given outer orientations to the sides and prescribed areas of the triangles formed by the origin with any two consecutive vertices if they exist such polygons are solutions to the discrete l 0 minkowski problem in the plane proposed in a larger generality by lutwak in his extension of the brunn minkowski theory the method exploits the equivalence between the l 0 polygons and the homothetic solutions to appropriate anisotropic crystalline flows hence as a by product we obtain a positive answer to jean taylor s conjecture on the uniqueness of polygonal shapes which are deformed homothetically under the flow by crystalline curvature without any symmetry assumptions

korteweg de vries burgers system in rn abstract we study cauchy problem in r n for the korteweg de vries burgers system parabolic regularization technique is used to prove its global in time solvability the regularization effect of the laplacian term is observed for the viscous solutions constructed in the paper asymptotic behavior of weak solutions is discussed next in the language of the global attractors

weakly damped forced korteweg de vries equations behave as a finite dimensional dynamical system in the long time abstract it is shown that a weak dissipation of order zero is sufficient to impose a finite dimensional behavior for korteweg de vries equations with periodic boundary conditions a universal attractor which captures all the trajectories is constructed its fractal dimension is proved to be finite

the lp busemann petty centroid inequality abstract the ratio between the volume of the p centroid body of a convex body k in r n and the volume of k attains its minimum value if and only if k is an origin symmetric ellipsoid this result the lp busemann petty centroid inequality was recently proved by lutwak yang and zhang in this paper we show that all the intrinsic volumes of the p centroid body of k are convex functions of a time like parameter when k is moved by shifting all the chords parallel to a fixed direction the lp busemann petty centroid inequality is a consequence of this general fact

perturbation theory of selfadjoint matrices and sign characteristics under generic structured rank one perturbations abstract for selfadjoint matrices in an indefinite inner product possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation genericity is understood in the sense of algebraic geometry special attention is paid to the perturbation behavior of the sign characteristic typically under such a perturbation for every given eigenvalue the largest jordan block of the eigenvalue is destroyed and in case the eigenvalue is real all other jordan blocks keep their sign characteristic the new eigenvalues i e those eigenvalues of the perturbed matrix that are not eigenvalues of the original matrix are typically simple and in some cases information is provided about their sign characteristic if the new eigenvalue is real the main results are proved by using the well known canonical forms of selfadjoint matrices in an indefinite inner product a version of the brunovsky canonical form and on general results concerning rank one perturbations obtained

moment entropy inequalities it is shown that the product of the renyi entropies of two independent random vectors provides a sharp lower bound for the expected value of the moments of the inner product of the random vectors this new inequality contains important geometry such as extensions of one of the fundamental affine isoperimetric inequalities the blaschke santalo inequality

valuations on manifolds and rumin cohomology smooth valuations on manifolds are studied by establishing a link with the rumin de rham complex of the co sphere bundle several operations on differential forms induce operations on smooth valuations signature operator rumin laplace operator euler verdier involution and derivation operator as an application alesker s hard lefschetz theorem for even translation invariant valuations on a finite dimensional euclidean space is generalized to all translation invariant valuations the proof uses kahler identities the rumin de rham complex and spectral geometry

global asymptotic stability for differentiable vector fields of r2 a let x r2 r2 be a differentiable map not necessarily c1 and let spec x be the set of complex eigenvalues of the derivative dxp when p varies in r2 if for some e 0 spec x 0 e then x is injective b let x r2 r2 be a differentiable vector field such that x 0 0 and spec x z c r z 0 then for all p r2 there is a unique positive trajectory starting at p moreover the limit set of p is equal to 0

rank one perturbations of h positive real matrices abstract we consider a generic rank one structured perturbation on h positive real matrices the case with complex rank one perturbation is treated in general but the main focus of this article is the real rank one perturbation in general the h positive real matrix a which is given in jordan canonical form loses the largest jordan block after a rank one perturbation for each eigenvalue surprisingly for a real h skew symmetric matrix for which the largest jordan block at eigenvalue zero has even size and for a real h nonnegative rank one perturbation the largest jordan block with zero eigenvalue grows one in size generic jordan structures of perturbed matrices are identified

a note on the strong convergence towards attractors of damped forced kdv equations abstract this note is a continuation of a previous article which appeared in this journal 1988 74 369 390 in the previous work we showed that a weak dissipation is suffiient to impose a finite dimensional behavior for korteweg de vries equations with space periodic boundary conditions more precisely it was shown that all the trajectories are captured by a finite dimensional attractor however we were only able to show that the convergence of the trajectories holds good in the sense of weak topology the goal of this note is to show that in fact this convergence holds true in the sense of strong topology

random attractor for stochastic reaction diffusion equation with multiplicative noise on unbounded domains abstract in this paper we study the asymptotic dynamics for stochastic reaction diffusion equation with multiplicative noise defined on unbounded domains we investigate the existence of a random attractor for the random dynamical system associated with the equation the asymptotic compactness of the random dynamical system is established by using uniform a priori estimates for far field values of solutions and a cut off technique

asymptotic behavior of a class of non autonomous degenerate parabolic equations abstract we investigate the asymptotic behavior of solutions of a class of non autonomous degenerate parabolic equations on r n with unbounded external forcing terms the existence of a pullback global attractor is proved in l 2 r n l p r n w 1 r r n for any p r 2

random attractors for stochastic reaction diffusion equations on unbounded domains the existence of a pullback attractor is established for a stochastic reaction diffusion equation on all n dimensional space the nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise the reaction diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far field values of solutions

asymptotic smoothing and the global attractor of a weakly damped kdv equation on the real line abstract the existence of the global attractor of a weakly damped forced korteweg de vries equation in the phase space l 2 r is proved an optimal asymptotic smoothing effect of the equation is also shown namely that for forces in l 2 r the global attractor in the phase space l 2 r is actually a compact set in h 3 r the energy equation method is used in conjunction with a suitable splitting of the solutions the dispersive regularization properties of the equation in the context of bourgain spaces are extensively exploited

a singular m matrix perturbed by a nonnegative rank one matrix has positive principal minors is it d stable abstract the positive stability and d stability of singular m matrices perturbed by non trivial nonnegative rank one perturbations is investigated in special cases positive stability or d stability can be established in full generality this is not the case as illustrated by a counterexample however matrices of the mentioned form are shown to be p matrices

projection bodies and valuations let be the projection operator which maps every polytope to its projection body it is well known that maps the set of polytopes pn in rn into pn that it is a valuation and that for every p pn p is affinely associated to p it is shown that these properties characterize the projection operator this proves a conjecture by lutwak

theory of valuations on manifolds iii multiplicative structure in the general case this is the third part of a series of articles where the theory of valuations on manifolds is constructed in the second part of this series the notion of a smooth valuation on a manifold was introduced the goal of this article is to put a canonical multiplicative structure on the space of smooth valuations on general manifolds thus extending some of the affine constructions from the first author s 2004 paper and from the first part of this series

an energy equation for the weakly damped driven nonlinear schro dinger equations and its application to their attractors abstract in this note we present an energy equation concerning the h 2 norm of a forced one dimensional nonlinear schrodinger equation as an application we are able to corroborate recent results on attractors of weakly damped forced nonlinear schrodinger equation by proving that the attractor is an attractor in the strong topology of h 2 and its hausdorff and fractal dimensions are finite in h 2 the energy equation for the h 1 norm can be derived in the same fashion and it implies corresponding results for the attractor in h 1

theory of valuations on manifolds ii abstract this article is the second part in the series of articles where we are developing theory of valuations on manifolds roughly speaking valuations could be thought as finitely additive measures on a class of nice subsets of a manifold which satisfy some additional assumptions the goal of this article is to introduce a notion of a smooth valuation on an arbitrary smooth manifold and establish some of the basic properties of it

random attractors for the stochastic fitzhugh nagumo system on unbounded domains abstract the existence of a random attractor for the stochastic fitzhugh nagumo system defined on an unbounded domain is established the pullback asymptotic compactness of the stochastic system is proved by uniform estimates on solutions for large space and time variables these estimates are obtained by a cut off technique

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

the global attractor for the 2d navier stokes flow on some unbounded domains we extend previous results on the existence of the global attractor for the 2d navier stokes equations on some unbounded domains in the sense that the forcing term need not lie in any weighted space nor is the boundary of the domain required to be smooth the existence of the global attractor is obtained on arbitrary open sets such that the poincare inequality holds and for forces in the natural dual space v the proof is based on the energy equation and the concept of asymptotic compactness an estimate of the hausdorff and fractal dimensions of the attractor is also given

the global attractor of the damped forced ostrovsky equation abstract the existence of the global attractor of the damped forced ostrovsky equation in l 2 r is proved for the forces in l 2 r moreover the global attractor of the equation in l 2 r is actually a compact set in h 3 r

general l_p affine isoperimetric inequalities sharp l p affine isoperimetric inequalities are established for the entire class of l p projection bodies and the entire class of l p centroid bodies these new inequalities strengthen the l p petty projection and the l p busemann petty centroid inequality

long time behavior for the weakly damped driven long wave short wave resonance equations abstract in this paper we consider the cauchy problem of the long wave short wave resonance equations by making use of a strichartz type inequality for the solutions decomposing suitably the solution semigroup into a decay parts and a more regular parts and ruling out the vanishing and dichotomy of the solutions we prove the existence of the global attractor and the asymptotic smoothing effect of the solutions

existence and finite dimensionality of the global attractor for evolution equations on unbounded domains abstract this paper is devoted to the asymptotic behavior for some evolution equations when the underlying euclidean domain is unbounded we present a method that is able to overcome the lack of compactness of the trajectories in such situations and that allows us to obtain some results on the global attractor similar to those on bounded domains this method is based on the utilization of an appropriate weight integration function that depends on x and t and becomes effective in x for large t s

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation