on the power method in max algebra abstract the eigenvalue problem for an irreducible nonnegative matrix a a ij in the max algebra system is a otimes x lambda x where a otimes x i mathop max j a ij x j and lambda turns out to be the maximum circuit geometric mean mu a a power method algorithm is given to compute mu a and eigenvector x the algorithm is developed by using results on the convergence of max powers of a which are proved using nonnegative matrix theory in contrast to an algorithm developed in 4 this new method works for any irreducible nonnegative a and calculates eigenvectors in a simpler and more efficient way some asymptotic formulas relating mu a the spectral radius and norms are also given

pmax1 and smax properties and asymptotic stability in the max algebra abstract in this work we introduce the class of p max 1 matrices for the max algebra and derive some properties of these that echo similar results for p matrices in the conventional algebra in analogy with 1 we define the row p max 1 property and the smax property of a finite set of nonnegative matrices moreover we relate these concepts to stability questions for sets of matrices and difference inclusions in the max algebra

a connection between state space and doubly coprime fractional representations explicit formulas for a doubly coprime fractional representation of the transfer function of a lumped lti system are given in terms of a stabilizable and detectable state space realization of the transfer function these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime factorizations several additional implications of this result are briefly discussed as are some immediate extensions and future research directions

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

on global existence and scattering for the wave maps equation we prove global existence and scattering for the wave maps equation in n 1 dimensions n 2 3 for initial data which is small in the scale invariant homogeneous besov space 2 1 n 2 x 2 1 n 2 1 this result was proved in an earlier paper by the author for n 4

barabanov norms lipschitz continuity and monotonicity for the max algebraic joint spectral radius abstract we present several results describing the interplay between the max algebraic joint spectral radius jsr for compact sets of matrices and suitably defined matrix norms in particular we extend a classical result for the conventional algebra showing that the max algebraic jsr can be described in terms of induced norms of the matrices in the set we also show that for a set generating an irreducible semigroup in a cone theoretic sense a monotone barabanov norm always exists this fact is then used to show that the max algebraic jsr is locally lipschitz continuous on the space of compact irreducible sets of matrices with respect to the hausdorff distance we then prove that the max algebraic jsr is locally hoelder continuous on the space of compact sets of nonnegative matrices finally we prove a strict monotonicity property for the max algebraic jsr that echoes a fact for the classical jsr the single matrix characterisation of the max algebraic jsr plays a vital role in our proofs

parameterizations of stabilizing compensators by using reduced order observers reduced order stabilizing compensators are parameterized in terms of doubly coprime factorizations having state space representations based on a minimal order observer and its dual observer a free parameter of the compensators is restricted within strictly proper stable rational functions the order of the compensators is lower than that of nett s by the number of outputs or inputs

generators extremals and bases of max cones abstract max cones are max algebraic analogs of convex cones in the present paper we develop a theory of generating sets and extremals of max cones in r n this theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors we give new proofs of existing results suitably generalizing restating and refining them of these it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements we include results on properties of open and closed cones on properties of totally dependent sets and on computational bounds for the problem of finding the essentially unique basis of a finitely generated cone

brief papers new results in state estimation and regulation an alternative technique to design linear state estimators and regulators is presented this technique is based on transfer matrix considerations the optimal regulator or estimator gain is obtained via spectral factorization and the solution of a simple equation in polynomial matrices this approach provides further insight displays the duality of estimation and control nicely and bridges the state space and frequency domain techniques the resulting design procedure is computationally attractive and particularly simple for system matrices in the observer or controller canonical form

almost optimal local well posedness for the 3 1 dimensional maxwell klein gordon equations this paper contains a detailed study of the local in time regularity properties of the maxwell klein gordon mkg equations the mkg equations represent a physical model for the interaction of a spin 0 particle sometimes referred to as a higgs or scalar field with the classical equations of electrodynamics i e maxwell s equations in terms of the calculus of variations the mkg system consists of the euler lagrange equations associated to the density l 4 f f 1 2 d d where f a a is the electro magnetic field associated to the set a of real valued potentials on the minkowski space d 1a is the corresponding covariant derivative with respect to those potentials and r c is a complex scalar field in terms of the field f the euler lagrange equations read f d 1a d d 0 1b

on prediction type reduced order observers for discrete time systems we define the combination of the standard reduced order observer with one step ahead state predictor as the prediction type reduced order observer we first obtain a simple and significant relation that ensures the equivalence of the reduced and the full order observers in the steady state we show the fact that a prediction type reduced order does not always provide a reduced order controller we also discuss the implications of the equivalence of the observers on state space representation of the doubly coprime factorization and yjb parametrization we use the equivalence relation to clarify the feedback properties achieved by the loop transfer recovery technique using the prediction type reduced order observer at both the input and output of a square and minimum phase plant in addition we discuss the computational requirements for the observer based controllers

local well posedness of the ostrovsky stepanyams and tsimring equation in sobolev spaces of negative indices abstract in this paper we prove by using the method of bilinear estimate in the bourgain type spaces that the initial value problem of the ost equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r when s 1

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

analyticity and smoothing effect for the korteweg de vries equation with a single point singularity we show that a solution of the cauchy problem for the kdv equation cases partial tv partial 3 x v partial x v 2 0 qquad t in t t x in bbb r v 0 x phi x endcases has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity at x 0 it is shown that for h s mathbb r s 3 4 data satisfying the condition sum k 0 infty frac a 0 k k x partial x k phi h s infty the solution is analytic in both space and time variable the above condition allows us to take as initial data the dirac delta measure or the cauchy principal value of 1 x the argument is based on the recent progress on the well posedness result by bourgain 2 and kenig ponce vega 20 and a systematic use of the dilation generator 3t partial t x partial x

modified doubly coprime fractional representations related to the reduced order observer modified doubly coprime fractional representations are introduced which are related to compensators based on reduced order observers from these representations the coprime compensator matrix fraction descriptions can directly be derived using standard software

an explicit lipschitz constant for the joint spectral radius abstract in 2002 wirth has proved that the joint spectral radius of irreducible compact sets of matrices is locally lipschitz continuous as a function of the matrix set in the paper an explicit formula for the related lipschitz constant is obtained

exact boundary controllability of the korteweg de vries equation we consider boundary control of the distributed parameter system described by the korteweg de vries kdv equation posed on a finite interval alpha leq x leq beta left begin array l u t u x uu x u xxx 0 u alpha t h 1 t qquad u beta t h 2 t qquad u x beta t h 3 t u end array right qquad for t geq 0 it is shown that by choosing appropriate control inputs h j t j 1 2 3 one can always guide the system from a given initial state phi in h s alpha beta to a given terminal state psi in h s alpha beta in the time period 0 t so long as phi and psi satisfy phi cdot w cdot 0 h s alpha beta leq delta quad mbox and quad psi cdot w cdot t h s alpha beta leq delta for some delta 0 independent of phi and psi where s geq 0 and w equiv w x t is a given smooth solution of the kdv equation this exact boundary controllability is established by considering a related initial value control problem of the kdv equation posed on the whole line r various recently discovered smoothing properties of the kdv equation have played important roles in our approach

well posedness of the cauchy problem for ostrovsky stepanyams and tsimring equation with low regularity data abstract in this paper we prove that the initial value problem of the ost equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r when s 3 4 and globally well posed in h s r when s 0

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

continuity of the generalized spectral radius in max algebra abstract let be an induced matrix norm associated with a monotone norm on r n and be the collection of all nonempty closed and bounded subsets of n n nonnegative matrices under this matrix norm for the hausdorff metric h between and is given by h max sup a inf b a b sup b inf a a b the max algebra system consists of the set of nonnegative numbers with sum a b max a b and the standard product ab for a b 0 for n n nonnegative matrices a b their product is denoted by a b where a b ij max 1 k n a ik b kj for each the max algebra version of the generalized spectral radius of is limsup m sup a m a 1 m where m a 1 a 2 a m a i here a is the maximum circuit geometric mean in this paper we prove that the max algebra version of the generalized spectral radius is continuous on the hausdorff metric space h the notion of the max algebra version of simultaneous nilpotence of matrices is also proposed necessary and sufficient conditions for the max algebra version of simultaneous nilpotence of matrices are presented as well

on the continuity of the generalized spectral radius in max algebra abstract given a bounded set of n n non negative matrices let and denote the generalized spectral radius of and its max version respectively we show that sup t 0 n 1 t 1 t where t denotes the hadamard power of we apply this result to give a new short proof of a known fact that is continuous on the hausdorff metric space h of all nonempty compact collections of n n non negative matrices

generalized spectral radius and its max algebra version abstract let c n n and r n n be bounded subsets and let and denote the generalized spectral radius of and the max algebra version of the generalized spectral radius of respectively we apply a single matrix description of to give a new elementary and straightforward proof of the berger wang formula in max algebra and consequently a new short proof of the original berger wang formula in the case of bounded subsets of n n non negative matrices we also obtain a new description of in terms of the schur hadamard product and prove new trace and max trace descriptions of and in particular we show that limsup m sup a m tr a 1 m limsup m sup a m tr a 1 m and limsup m sup b m tr b 1 m limsup m sup b m tr b 1 m where tr a max i 1 n a ii and b b ij

parameterization of reduced order compensator in the frequency domain after a brief survey of the relations between time and frequency domain representations of the observer based linear state feedback loop the parameterization of the compensator is discussed it is shown that the free parameters specified by the state feedback matrix k and the output error feedback matrix l in the time domain appear as coefficients in the polynomial matrices dtilde s and s characterizing the closed loop dynamics in the frequency domain these polynomial matrices can either be chosen arbitrarily pole placement problem or they may be derived as solutions from the corresponding optimal control or the optimal estimation problems respectively starting from the polynomial matrices dtilde s and s the doubly coprime fractional representations of the system and of the compensator transfer matrices can be computed directly without recurrence to the time domain results from these representations the right and left coprime compensator matrix fraction descriptions can easily be evalu

max plus definite matrix closures and their eigenspaces in this paper we introduce the definite closure operation for matrices with finite permanent reveal inner structures of definite eigenspaces and establish some facts about hilbert distances between these inner structures and the boundary of the definite eigenspace

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

well posedness for the stochastic higher order modified camassa holm equation abstract in this article the stochastic higher order modified camassa holm equation with additive noise is considered using the fourier restriction norm method and bilinear estimates the local existence and uniqueness result is proved by the contraction mapping principle the global existence of the h 1 r solution of this problem is also obtained by energy inequality

extremal norms for positive linear inclusions abstract for finite dimensional linear semigroups which leave a proper cone invariant it is shown that irreducibility with respect to the cone implies the existence of an extremal norm in case the cone is simplicial a similar statement applies to absolute norms the semigroups under consideration may be generated by discrete time systems continuous time systems or continuous time systems with jumps the existence of extremal norms is used to extend results on the lipschitz continuity of the joint spectral radius beyond the known case of semigroups that are irreducible in the representation theory interpretation of the word

strictly doubly coprime factorizations and all stabilizing compensators related to reduced order observers abstract doubly coprime factorizations dcf play an important role in the design of compensators for lumped linear time invariant systems they are defined as stable proper rational matrices and they not only constitute a starting point for the fractional approach to all stabilizing compensators but they are also closely related to the frequency domain design of optimal state feedback and estimation and of the resulting observer based compensators when defining dcfs related to reduced order observers some rational matrices become improper this can be prevented by the introduction of artificial stable dynamics the identity elements which cancel in the system and in the compensator transfer matrices such pole zero cancellations however are contrary to the usual notion of coprimeness it is shown here that these additional dynamics which have no meaning in an observer based compensator scheme are superfluous and that the parametrization of all stabilizing compensators is feasible on the basis of possibly nonproper factorizations not containing identity elements for such factorizations the new definition of strictly doubly coprime factorizations sdcf is proposed

the generalized spectral radius and extremal norms the generalized spectral radius also known under the name of joint spectral radius or after taking logarithms maximal lyapunov exponent of a discrete inclusion is examined we present a new proof for a result of barabanov which states that for irreducible sets of matrices an extremal norm always exists this approach lends itself easily to the analysis of further properties of the generalized spectral radius we prove that the generalized spectral radius is locally lipschitz continuous on the space of compact irreducible sets of matrices and show a strict monotonicity property of the generalized spectral radius sufficient conditions for the existence of extremal norms are obtained

a max version of the generalized spectral radius theorem abstract let be a bounded set of n n nonnegative matrices in max algebra in this paper we propose the notions of the max algebra version of the generalized spectral radius of and the max algebra version of the joint spectral radius of the max algebra version of the generalized spectral radius theorem is established we propose the relationship between the generalized spectral radius of in the sense of daubechies and lagarias and its max algebra version moreover a generalization of elsner and van den driessche s lemma is presented as well

doubly coprime matrix fraction representations using proportional and derivative feedback concepts in generalized state space systems the concepts of proportional and derivative state feedback are used to derive some explicit formulas for doubly coprime matrix fraction representations of generalized state space systems specifically proportional and derivative state feedback is used to determine the doubly coprime matrix fraction representations and to solve the corresponding generalized bezout identity in polynomial matrix form

well posedness and ill posedness of kdv equation with higher dispersion abstract first the cauchy problem for kdv equation with 2 n 1 order dispersion is studied and the local well posedness result for the initial data in sobolev spaces h s r with s n 1 4 is established via the fourier restriction norm method second we prove that the kdv equation with 2 n 1 order dispersion is ill posed for the initial data in h s r with s n 1 4 n 2 n n if the flow map is c 2 differentiable at zero form h s r to c 0 t h s r finally we obtain the sharp regularity requirement for the kdv equation with 2 n 1 order dispersion s n 1 4

the finiteness conjecture for the generalized spectral radius of a set of matrices the generalized spectral radius g9 of a set of n n matrices is g9 lim supk g9k 1k where g9k sup a1a2 ak each ai the joint spectral radius g9 is g9 lim supk g9k 1k where g9k sup a1 ak each ai it is known that g9 g9 holds for any finite set of n n matrices the finiteness conjecture asserts that for any finite set of real n n matrices there exists a finite k such that g9 g9 g9k 1k the normed finiteness conjecture for a given operator norm asserts that for any finite set a1 am having all ai op 1 either g9 1 or g9 g9 g9k 1k 1 for some finite k it is shown that the finiteness conjecture is true if and only if the normed finiteness conjecture is true for all operator norms the normed finiteness conjecture is proved for a large class of operator norms extending results of gurvits in particular for polytope norms and for the euclidean norm explicit upper bounds are given for the least k having g9 g9k 1k these results imply upper bounds for generalized critical exponents for these norms

performance evaluation of max automata automata with multiplicities over the max semiring can be used to represent the behavior of timed discrete event systems this formalism which extends both conventional automata and max linear representations covers a class of systems with synchronization phenomena and variable schedules performance evaluation is considered in the worst mean and optimal cases a simple algebraic reduction is provided for the worst case the last two cases are solved for the subclass of deterministic series recognized by deterministic automata deterministic series frequently arise due to the finiteness properties of max linear projective semigroups the mean performance is given by the kolmogorov equation of a markov chain the optimal performance is given by a hamilton jacobi bellman equation

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

the minkowski theorem for max plus convex sets abstract we establish the following max plus analogue of minkowski s theorem any point of a compact max plus convex subset of r n can be written as the max plus convex combination of at most n 1 of the extreme points of this subset we establish related results for closed max plus cones and closed unbounded max plus convex sets in particular we show that a closed max plus convex set can be decomposed as a max plus sum of its recession cone and of the max plus convex hull of its extreme points

the generalized spectral radius is strictly increasing using a result linking convexity and irreducibility of matrix sets it is shown that the generalized spectral radius of a compact set of matrices is a strictly increasing function of the set in a very natural sense as an application some consequences of this property in the area of time varying stability radii are discussed in particular using the implicit function theorem sufficient conditions for lipschitz continuity are derived an example is presented of a linearly increasing family of matrix polytopes for which the proximal subgradient of the generalized spectral radius at a certain polytope contains 0 so that the implicit function theorem is not applicable in all cases

on the max version of the generalized spectral radius theorem abstract let be a bounded set of n n non negative matrices recently the max algebra version of the generalized spectral radius of was introduced we show that lim t i t 1 t where denotes the generalized spectral radius and t the hadamard power of this provides a description of that uses no max terminology as an application we give a short proof of the max version of the generalized spectral radius theorem

a max version of the perron frobenius theorem abstract if a an n n nonnegative irreducible matrix then there exists a 0 and a positive vector x such that maxjaijxj a xi i 1 2 n furthermore a is the maximum geometric mean of a circuit in the weighted directed graph corresponding to a this theorem which we refer to as the max version of the perron frobenius theorem is well known in the context of matrices over the max algebra and also in the context of matrix scalings in the present work which is partly expository we bring out the intimate connection between this result and the perron frobenius theory we present several proofs of the result some of which use the perron frobenius theorem structure of max eigenvalues and max eigenvectors is described possible ways to unify the perron frobenius theorem and its max version are indicated some inequalities for a are proved

a nonminimal representation of reduced order observers abstract a new nonminimal representation of reduced order observers of order n k in the time and in the frequency domain is introduced this reduced order observer is based on a full order plant model where the order reduction to n k results from an infinite gain feedback of k outputs this new representation is crucial for establishing the links between time and frequency domain design methods incorporating reduced order observers the main result are the doubly coprime fractional representations of a transfer matrix related to an observer of order n k

strichartz estimates for dispersive equations and solvability of the kawahara equation abstract we first establish a series of strichartz estimates for a general class of linear dispersive equations by applying the theory of oscillatory integrals established by kenig ponce and vega next we use such estimates to study solvability of the cauchy problem of the kawahara equation t u a u x u x 3 u x 5 u 0 in the class c r h s r local existence is proved for s 1 4 and global existence is proved for s 2

asymptotics frequency modulation and low regularity ill posedness for canonical defocusing equations in a recent paper kenig ponce and vega study the low regularity behavior of the focusing nonlinear schrodinger nls focusing modified korteweg de vries mkdv and complex korteweg de vries kdv equations using soliton and breather solutions they demonstrate the lack of local well posedness for these equations below their respective endpoint regularities in this paper we study the defocusing analogues of these equations namely defocusing nls defocusing mkdv and real kdv all in one spatial dimension for which suitable soliton and breather solutions are unavailable we construct for each of these equations classes of modified scattering solutions which exist globally in time and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts these solutions are used to demonstrate lack of local well posedness in certain sobolev spaces in the sense that the dependence of solutions upon initial data fails to be uniformly continuous in particular we show that the mkdv flow is not uniformly continuous in the l 2 topology despite the existence of global weak solutions at this regularity finally we investigate the kdv equation at the endpoint regularity h and construct solutions for both the real and complex kdv equations the construction provides a nontrivial time interval t t and a locally lipschitz continuous map taking the initial data in h to a distributional solution u c 0 t t h which is uniquely defined for all smooth data the proof uses a generalized miura transform to transfer the existing endpoint regularity theory for mkdv to kdv

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

refined energy inequality with application to well posedness for the fourth order nonlinear schrodinger type equation on torus we consider the time local and global well posedness for the fourth order nonlinear schrodinger type equation 4nls on the torus the nonlinear term of 4nls contains the derivatives of unknown function and this prevents us to apply the classical energy method to overcome this difficulty we introduce the modified energy and derive an a priori estimate for the solution to 4nls

a new approach for calculating doubly coprime matrix fraction descriptions using the concepts of state feedback in generalized state space explicit formulas for calculating doubly coprime matrix fraction descriptions mfds of the transfer matrix of a regular state space system and the polynomial generalized bezout identity elements corresponding to those coprime mfds are presented they can be easily calculated with the help of existing computational algorithms and software packages

doubly coprime factorizations reduced order observers and dynamic state estimate feedback doubly coprime factorizations of the transfer function of a lumped linear time invariant system are a starting point for many of the results in the factorization approach to multivariabie control system analysis and synthesis in work by nett et al 1984 explicit state space realizations of these factorizations are derived using results from state estimation state feedback theory here new doubly coprime factorizations are developed based on minimal order observers following on from this various extensions are noted and it is proved that the class of all proper stabilizing controllers for a given plant can be generated by dynamic feedback of the reduced order state estimate

global solutions of the klein gordon schrodinger system with rough data in r 2 1 abstract in this paper we consider the klein gordon schrodinger system with quadratic yakuwa coupling and cubic autointeractions in r 2 1 and prove the existence and uniqueness of global solution for rough data the techniques to be used are adapted from a general scheme originally introduced by j bourgain to split the data into the low and high frequency parts

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