well posedness for the schrodinger korteweg de vries system we study well posedness of the cauchy problem associated to the schrodinger korteweg de vries system we obtain local well posedness for weak initial data where the best result obtained is for data in the sobolev space l 2 r x h 3 4 this result implies in particular the global well posedness in the energy space h 1 r x h 1 r both results considerably improve the previous ones by bekiranov ogawa and ponce 1997 guo and miao 1999 and tsutsumi 1993

counter synchronization using the thue morse sequence and psk the thue morse sequence tms the parity check sequence formed on a normal binary counter has been proposed as a vehicle for comma free synchronizatin of binary counters this note reports further work on the tms as a vehicle for commafree synchronization of binary counters specifically we present a new synchronization algorithm and a way to implement the algorithm via psk encoding dpsk decoding and we examine and comment on the tms bpsk spectrum

partial period crosscorrelations of geometric sequences for a binary pseudorandom sequence s sub i with period n the partial period autocorrelation function a sub s spl tau k d is defined by correlating the portion of the sequence within a window of size d and start position k with the portion in another window of the same size but starting spl tau steps later in the sequence a distribution of possible partial period autocorrelation values is obtained by allowing the start position k to vary over all possible values o spl les k

on the periodic schrodinger boussinesq system abstract we study the local and global well posedness of the periodic boundary value problem for the nonlinear schrodinger boussinesq system the existence of periodic traveling wave solutions as well as the orbital stability of such solutions are also considered

the merit factor of legendre sequences corresp turyn has shown by numerical computation that the merit factor of long legendre sequences offset by a quarter of their length has in all likelihood the asymptotic value 6 the ergodicity postulate is used in this correspondence to calculate that the merit factor f of a legendre sequence offset by a fraction f of its length has an asymptotic value given by 1 f 2 3 4 f 8f 2 f leq 1 2 which gives f 6 for f 1 4

spectral properties of the thue morse sequence the thue morse sequence tms has been proposed as a vehicle for effecting comma free synchronization the purpose of this correspondence is twofold first we present and examine an expression which describes the tms power spectral density second we consider the autocorrelation of the tms sequence and cast it as a random process

interaction equations for short and long dispersive waves abstract we show the time local well posedness for a system of nonlinear dispersive equations for the water wave interaction formula it is shown that for any initial data u0 v0 hs r hs 1 2 r s 0 the solution for the above equation uniquely exists in a subset ofc t t hs c t t hs 1 2 and depends continuously on the data by virtue of a special structure of the nonlinear coupling the solution is stable under a singular limiting process

multilinear estimates for periodic kdv equations and applications we prove an endpoint multilinear estimate for the xs b spaces associated to the periodic airy equation as a consequence we obtain sharp local well posedness results for periodic generalized kdv equations as well as some global well posedness results below the energy norm

the peak sidelobe level of families of binary sequences a numerical investigation is presented for the peak sidelobe level psl of legendre sequences maximal length shift register sequences m sequences and rudin shapiro sequences the psl gives an alternative to the merit factor for measuring the collective smallness of the aperiodic autocorrelations of a binary sequence the growth of the psl of these infinite families of binary sequences is tested against the desired growth rate o spl radic nlnn for sequence length n the claim that the psl of m sequences grows like o spl radic n which appears frequently in the radar literature is concluded to be unproven and not currently supported by data notable similarities are uncovered between the psl and merit factor behavior under cyclic rotations of the sequences

generalized strichartz estimates and scattering for 3d zakharov system we obtain scattering for the 3d zakharov system with non radial small data in the energy space with angular regularity of degree one the main ingredient is a generalized strichartz estimate for the schrodinger equation in the space of l2 angular integrability

rudin shapiro like sequences with maximum asymptotic merit factor borwein and mossinghoff investigated the rudin shapiro like sequences which are infinite families of binary sequences usually represented as polynomials each family of rudin shapiro like sequences is obtained from a starting sequence which we call the seed by a recursive construction that doubles the length of the sequence at each step and many sequences produced in this manner have exceptionally low aperiodic autocorrelation borwein and mossinghoff showed that the asymptotic autocorrelation merit factor for any such family is at most 3 and found the seeds of length 40 or less that produce the maximum asymptotic merit factor of 3 the definition of rudin shapiro like sequences was generalized by katz lee and trunov to include sequences with arbitrary complex coefficients among which are families of low autocorrelation polyphase sequences katz lee and trunov proved that the maximum asymptotic merit factor is also 3 for this larger class here we show that a family of such rudin shapiro like sequences achieves asymptotic merit factor 3 if and only if the seed is either of length 1 or is the interleaving of a pair of golay complementary sequences for small seed lengths where this is not possible the optimal seeds are interleavings of pairs that are as close as possible to being complementary pairs and the idea of an almost complementary pair makes sense of remarkable patterns in previously unexplained data on optimal seeds for binary rudin shapiro like sequences

a new signal representation a new method is presented for representing and approximating signals which are made up of odd frequency components the method uses a half basis set all of whose members exhibit identical power spectral densities in contrast to the conventional fourier approximation approximations by this new method will spread the approximation error over the odd frequencies in the half basis approximations to the full set of signals i e those containing both even and odd frequency components can also be realized by this new method

local well posedness for the sixth order boussinesq equation abstract in this work we study the local well posedness of the initial value problem for the nonlinear sixth order boussinesq equation u t t u x x u x x x x u x x x x x x u 2 x x where 1 we prove the local well posedness with initial data in non homogeneous sobolev spaces h s r for negative indices of s r

necessary and sufficient conditions for the two point phase probability function of two phase random media constructing realizations of random media with a specified two point phase probability function s2 has attracted considerable attention in the recent literature however little is known about cond

bilinear space time estimates for linearised kp type equations on the three dimensional torus with applications abstract a bilinear estimate in terms of bourgain spaces associated with a linearised kadomtsev petviashvili type equation on the three dimensional torus is shown as a consequence time localized linear and bilinear space time estimates for this equation are obtained applications to the local and global well posedness of dispersion generalised kp ii equations are discussed especially it is proved that the periodic boundary value problem for the original kp ii equation is locally well posed for data in the anisotropic sobolev spaces h x s h y e t 3 if s 1 2 and e 0

the merit factor of binary sequences related to difference sets long binary sequences related to cyclic difference sets are investigated among all known constructions of cyclic difference sets it is shown that only sequences constructed from hadamard difference sets can have an asymptotic nonzero merit factor maximal length shift register sequences legendre and twin prime sequences are all constructed from hadamard difference sets the authors prove that the asymptotic merit factor of any maximal length shift register sequence is three for twin prime sequences it is shown that the best asymptotic merit factor is six this value is obtained by shifting the twin prime sequence one quarter of its length it turns out that legendre sequences and twin prime sequences have similar behavior jacobi sequences are investigated on the basis of the jacobi symbol the best asymptotic merit factor is shown to be six through the introduction of product sequences it is argued that the maximal merit factor among all sequences of length n is at least six when n is large the authors also demonstrate that it is fairly easy to construct sequences of moderate composite length with a merit factor close to six

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

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

an improved local well posedness result for the one dimensional zakharov system abstract the 1d cauchy problem for the zakharov system is shown to be locally well posed for low regularity schrodinger data u 0 h k p and wave data n 0 n 1 h l p h l 1 p under certain assumptions on the parameters k l and 1 p 2 where u 0 h k p k u 0 l p generalizing the results for p 2 by ginibre tsutsumi and velo especially we are able to improve the results from the scaling point of view and also allow suitable k 0 l 1 2 i e data u 0 l 2 and n 0 n 1 h 1 2 h 3 2 which was excluded in the case p 2

multivalued fredholm type operators with abstract generalised inverses multivalued semi fredholm type operators with topologically complemented kernels and ranges are investigated in normed linear spaces it is shown that regularisers generalised inverses can be constructed for the classes given and that the operators considered can be characterised in terms of their regularisers continuity of the inverses is discussed and dual properties of adjoints are given

asymptotic smoothing effect for a weakly damped nonlinear schrodinger equation in t2 abstract we prove that the global attractor for a weakly damped two dimensional nonlinear schrodinger equation in the usual energy space is in fact included and compact in a more regular energy space the method relies on a suitable approximation when time goes to infinity of the high frequency modes of the solutions

sharp local well posedness for the good boussinesq equation abstract in the present article we prove the sharp local well posedness and ill posedness results for the good boussinesq equation on 1d torus the initial value problem is locally well posed in h 1 2 t and ill posed in h s t for s 1 2 well posedness result is obtained from reduction of the problem into a quadratic nonlinear schrodinger equation and the contraction argument in suitably modified x s b spaces the proof of the crucial bilinear estimates in these spaces especially in the lowest regularity rely on some bilinear estimates for periodic functions in x s b spaces which are generalization of the bilinear refinement of the l 4 strichartz estimate on r our result improves the known local well posedness in h s t with s 3 8 given by oh stefanov 2012 similar ideas also establish the sharp local well posedness in h 1 2 r and ill posedness below h 1 2 for the nonperiodic case which improves the result of tsugawa and the author 2010 in h s r with s 1 2

well posedness for the cauchy problem associated to a periodic boussinesq equation abstract we consider the initial value problem associated to the nonlinear periodic sixth order boussinesq equation for given initial data in sobolev spaces h s t the local well posedness of this equation is established for s 1 2

low regularity solutions of non homogeneous boundary value problems of a higher order boussinesq equation in a quarter plane abstract in this article we study an initial boundary value problem of the sixth order boussinesq equation on a half line with nonhomogeneous boundary conditions u t t u x x u x x x x u x x x x x x u 2 x x 0 x 0 t 0 u x 0 x u t x 0 x x 0 u 0 t h 1 t u x x 0 t h 2 t u x x x x 0 t h 3 t t 0 where 1 it is shown that the problem is locally well posed in c r l o c h s r with initial condition h s r h s 1 r and boundary condition h 1 h 2 h 3 in the product space h s 1 3 r h s 1 3 r h s 3 3 r for 1 2 s 0

low regularity solutions for a 2d quadratic nonlinear schrodinger equation we establish that the initial value problem for the quadratic nonlinear schrodinger equation iu t u u 2 where u r 2 x r c is locally well posed in h s r 2 when s 1 the critical exponent for this problem is s c 1 and previous work by colliander delort kenig and staffilani 2001 established local well posedness for s 3 4

on the zakharov and zakharov schulman systems abstract we consider the initial value problem for the zakharov system formula which models the long wave langmuir turbulence in a plasma using the standard iteration scheme in the original system it is shown that the system is locally well posed uniformly on the ionic speed of sound in appropriate sobolev spaces the method provides similar results for the ivp for the zakharov schulman system formula where u r d 0 c r d 0 r and formula with a i j k a k i j real constants which models the interactions of small amplitude high frequency waves with acoustic type waves

characterizing the autocorrelations of binary sequences the class of autocorrelation functions of binary sequences is studied in terms of their finite dimensional projections that is the vector ranges of finite sets of consecutive correlation coefficients these are convex polyhedra whose extreme points can be generated by binary sequences of bounded periods algorithms are given for the calculation of the extreme points tables are presented for the extreme points and supporting hyperplanes of projections of dimensions up to five some conclusions about the general class of autocorrelations of binary sequences are drawn from these tables

well posedness for a quadratic derivative nonlinear schrodinger system at the critical regularity abstract we consider the cauchy problem for a quadratic derivative nonlinear schrodinger system in sobolev space h s in this paper under the null condition and the mass resonance relation we prove large data local well posedness if the space dimension d 3 and s s c d 2 and s s c or d 1 and s 0 where s c is the scaling critical regularity moreover under the same assumptions we also prove small data global well posedness and scattering in h s if d 3 and s s c our proof is based on a contraction argument using the u p and v p type function spaces

autocorrelation properties of a class of infinite binary sequences corresp a class of infinite pm 1 sequences is presented whose autocorrelation function is zero for all nonzero shifts

on the existence of a maximizer for the strichartz inequality it is shown that a maximizing function u l 2 does exist for the strichartz inequality e it x 2 u l 6 t l 6 x s u l 2 with s 0 being the sharp constant

sharp well posedness and ill posedness results for a quadratic non linear schrodinger equation abstract we establish that the quadratic non linear schrodinger equation iu t u xx u 2 where u r r c is locally well posed in h s r when s 1 and ill posed when s 1 previous work in c kenig g ponce l vega quadratic forms for the 1 d semilinear schrodinger equation trans amer math soc 346 1996 3323 3353 had established local well posedness for s 3 4 the local well posedness is achieved by an iteration using a modification of the standard x s b spaces the ill posedness uses an abstract and general argument relying on the high to low frequency cascade present in the non linearity and a computation of the first non linear iterate

on schrodinger maps we study the question of well posedness of the cauchy problem for schrodinger maps from 2 to the sphere 2 or to 2 the hyperbolic space the idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear schrodinger system of equations and then study this modified schrodinger map system msm we then prove local well posedness of the cauchy problem for the msm with minimal regularity assumptions on the data and outline a method to derive well posedness of the schrodinger map itself from it in proving well posedness of the msm the heart of the matter is resolved by considering truly quatrilinear forms of weighted l2 functions 2002 wiley periodicals inc

the cauchy problem for nonlinear quadratic interactions of the schrodinger type in one dimensional space in this work we study the well posedness of the cauchy problem associated with the coupled schrodinger equations with quadratic nonlinearities which appears modeling problems in nonlinear optics we obtain the local well posedness for data in sobolev spaces with low regularity to obtain the local theory we prove new bilinear estimates for the coupling terms of the system in the continuous case concerning global results in the continuous case we establish the global well posedness in hs r hs r for some negatives indexes s the proof of our global result uses the i method introduced by colliander et al

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

some non existence results for the semilinear schrodinger equation without gauge invariance we consider the cauchy problem for the semilinear schrodinger equation nls i t u u p t x 0 t rd u 0 x f x x rd where u u t x is a c valued unknown function c 0 p 1 0 f f x is a c valued given function and t is a maximal existence time of the solution our first aim in the present paper is to prove a large data blow up result for nls in hs critical or hs subcritical case p ps 1 4 d 2s for some s 0 more precisely we show that in the case 1 0 such that for any 0 the following estimates t c and limt t 0 u t hs if1 p ps u ltr 0 t b 2s ifp ps hold where c 0 are constants independent of and r is an admissible pair see theorem 2 3 our second aim is to prove non existence local weak solution for nls in the hs supercritical case p ps which means that if p ps then there exists an hs function f such that if there exist t 0 and a weak solution u to nls on 0 t then 0 see theorem 2 5

low regularity global well posedness for the zakharov and klein gordon schrodinger systems we prove low regularity global well posedness for the id zakharov system and the 3d klein gordon schrodinger system which are systems in two variables u r d x x r t c and n r d x x r t r the zakharov system is known to be locally well posed in u n l 2 x h and the klein gordon schrodinger system is known to be locally well posed in u n l 2 x l 2 here we show that the zakharov and klein gordon schrodinger systems are globally well posed in these spaces respectively by using an available conservation law for the l 2 norm of u and controlling the growth of n via the estimates in the local theory

improved local well posedness for the periodic good boussinesq equation abstract we prove that the good boussinesq model is locally well posed in the space h h 2 3 8 in the proof we employ the method of normal forms which allows us to explicitly extract the rougher part of the solution while we show that the remainder is in the smoother space c 0 t h t min 1 3 1 2 thus we establish a smoothing effect of order min 1 2 1 2 for the nonlinear evolution this is new even in the previously considered cases 0 1 4

autocorrelation properties of the thue morse sequence and their use in synchronization further properties of the autocorrelation coefficients of the thue morse sequence are developed which are then used to develop a novel robust synchronization method its behavior is simulated over a highly degraded binary symmetric channel and uses for the synchronization method are proposed the synchronization method has the potential to maintain synchronization at bit error rates up to 10

a perturbation result for q open quotient morphisms in normed spaces and applications to linear relations using some techniques of perturbation theory for quotient morphisms between banach spaces we obtain necessary and sufficient conditions for the stability of the topological index of a q open quotient morphism under small with respect to the gap topology perturbations with quotient morphisms as an application we obtain necessary and sufficient conditions for the stability of the topological index of an open linear relation with closed multivalued part between normed spaces under small perturbations with linear relations

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

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

well posedness and nonlinear smoothing for the good boussinesq equation on the half line abstract in this paper we study the regularity properties of the good boussinesq equation on the half line we obtain local existence uniqueness and continuous dependence on initial data in low regularity spaces moreover we prove that the nonlinear part of the solution on the half line is smoother than the initial data obtaining half derivative smoothing of the nonlinear term in some cases our paper improves the result in 17 being the first result that constructs solutions for the initial and boundary value problem of the good boussinesq equation below the l 2 space our theorems are sharp within the framework of the restricted norm method that we use and match the known results on the full line in 20 and 13

global dynamics below the ground state energy for the zakharov system in the 3d radial case abstract we consider the global dynamics below the ground state energy for the zakharov system in the 3d radial case we obtain dichotomy between the scattering and the growup

the cauchy problem for the dissipation modified kadomtsev petviashvili equation abstract considered herein is the dissipation modified kadomtsev petviashvili equation in two space dimensional case it is established that the cauchy problem associated to this equation is locally well posed in anisotropic sobolev spaces it is also shown in some sense that this result is sharp in addition the global well posedness for this equation under suitable conditions is proved

ill posedness for a system of quadratic nonlinear schrodinger equations in two dimensions abstract this paper concerns the ill posedness issue for a system of quadratic nonlinear schrodinger equations in two dimensions from previous studies for the large time behavior of the solution to the system one may expect that the critical regularity to show the well posedness depends on the dispersion coefficients to prove the ill posedness we show the failure of the uniform continuity of the data solution map for the mass resonance case and show the norm inflation for other cases

local and global well posedness for the 2d zakharov kuznetsov burgers equation in low regularity sobolev space abstract in the present paper we consider the cauchy problem of the 2d zakharov kuznetsov burgers zkb equation which has the dissipative term x 2 u this is known that the 2d zakharov kuznetsov equation is well posed in h s r 2 for s 1 2 and the 2d nonlinear parabolic equation with quadratic derivative nonlinearity is well posed in h s r 2 for s 0 by using the fourier restriction norm with dissipative effect we prove the well posedness for zkb equation in h s r 2 for s 1 2

low regularity bilinear estimates for a quadratic nonlinear schrodinger equation abstract in this article we establish the bilinear estimates corresponding to the 1d and 2d nls with a quadratic nonlinearity c 1 u 2 c 2 u 2 which imply the local well posedness of the cauchy problem in h s for s 1 in the 1d case and for s 1 in the 2d case this is a continuation of our study n kishimoto local well posedness for the cauchy problem of the quadratic schrodinger equation with nonlinearity u 2 commun pure appl anal 7 2008 1123 1143 on the 1d nls with nonlinearity c u 2 previous papers by kenig ponce and vega and colliander delort kenig and staffilani established local well posedness for s 3 4 in 1d and in 2d respectively and when the nonlinearity is restricted to c u 2 papers by bejenaru and tao and bejenaru and de silva improved these results to s 1 in 1d and s 1 in 2d the bilinear estimate for 2d also yields an improvement on the growth rate of sobolev norms of finite energy global in time solutions to the 2d cubic nls

stability of the index of a semi fredholm complex of banach spaces abstract we prove the stability of the index and the semicontinuity of the dimensions of the cohomology groups of semi fredholm complexes of banach spaces and closed linear operators with respect to perturbations of the operators and of the underlying spaces which are small with respect to the gap topologies it seems that even for single semi fredholm operators some of the statements are more general than the current ones the results are applied to obtain semicontinuity for joint spectra of finite systems of commuting bounded linear operators

equations of langmuir turbulence and nonlinear schrodinger equation smoothness and approximation we consider the following family of systems parametrized by e 0 which describe langmuir s turbulence in the case where initial data are sufficiently small we study the asymptotic behaviour of the solutions ee ne when e goes to zero for d 1 2 3 namely we state convergence results of ee ne to the couple e e 2 where e is the solution of the nonlinear schrodinger equation moreover in three dimensions we show that the smooth solution ee ne is defined on 0 tmax e with tmax e going to infinity as e goes to 0

aperiodic correlations and the merit factor of a class of binary sequences corresp a class of binary sequences of length n 2 m is considered and it is shown that their aperiodic autocorrelations can be calculated recursively in a simple way based on this the merit factor of the sequences is calculated and it is shown that the asymptotic value is 3 finally it is proved that the magnitude of the maximal aperiodic autocorrelation is bounded by n 0 9

quadratic dispersive generalized benjamin ono equation abstract this paper is concerned with quadratic dispersive generalized benjamin ono equation 1 1 1 2 we prove sharp local well posedness in h s s 1 4 with a solution map that is analytic from h s r to c 0 t h s r and the ill posed result in h s r as soon as s s in the sense that the flow map u 0 u t cannot be continuous

global solutions of the derivative schrodinger equation in a class of functions analytic in a strip lower bounds on the rate of decrease in time of a uniform radius of spatial analyticity for solutions of the derivative schrodinger equation are derived the bounds depend algebraically on time they are valid as long as the initial datum is of order one in l2 norm and satisfies suitable spatial decay requirements on the real axis

solvability of the cauchy problem of nonlinear beam equations in besov spaces abstract in this paper we study solvability of the cauchy problem of the nonlinear beam equation t 2 u 2 u u p with initial data in besov spaces we prove that for any 1 q the cauchy problem of this equation is locally well posed in the besov spaces b 2 q s p r n and b 2 q s r n where s p n 2 4 p 1 and s s p and globally well posed in these spaces if initial data are small moreover we obtain scattering results in b 2 q s p r n

quadratic forms for the 1 d semilinear schrodinger equation this paper is concerned with 1 d quadratic semilinear schr6dinger equations we study local well posedness in classical sobolev space hs of the associated initial value problem and periodic boundary value problem our main interest is to obtain the lowest value of s which guarantees the desired local well posedness result we prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered

on the covariance function of stationary binary sequences with given mean recent work on characterizing the covariance functions of stationary binary sequences has been geometric in flavor extreme distributions induced in windows of a given length are obtained from special periodic sequences the work is extended to sequences with a given mean m revealing some finer structure

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

on well posedness for the benjamin ono equation we prove existence and uniqueness of solutions for the benjamin ono equation with data in h s mathbb r s 1 4 moreover the flow is holder continuous in weaker topologies