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

almost sure well posedness of the cubic nonlinear schrodinger equation below l 2 mathbb t we consider the cauchy problem for the one dimensional periodic cubic non linear schrodinger equation nls with initial data below l 2 in particular we exhibit nonlinear smoothing when the initial data are randomized then we prove local well posedness of nls almost surely for the initial data in the support of the canonical gaussian measures on h s t for each s 1 and global well posedness for each s 1 12

some remarks on the blow up rate for the 3d magnetic zakharov system we continue our study 8 on the cauchy problem for the three dimensional magnetic zakharov system zsm iet e ne i e b e 0 nt v vt n e 2 b i e e b 0 with initial data zsm 1 e 0 x e0 x n 0 x n0 x v 0 x v0 x let e n v c 0 t h1 r3 l2 r3 l2 r3 be a blow up solution to the cauchy problem zsm zsm 1 and let t be its blow up time then e n satisfies the space time integral estimate e 1 0t r3 n t x qdx 1q r3 e t x 2qdx 1q dt where 1e e 0 14 and q 32 1 e 32 2 the estimate e 1 implies that for a 1 supt 0 t t t ae n t lq r3 e t l2q r3 2 in particular if e n v is a radially symmetric solution to zsm zsm 1 then supt 0 t t t ae n t lq d e t l2q d 2 where a 0 13 and d x r3 x 1

control and stabilization of the benjamin ono equation in l 2 mathbb t we study the control and stabilization of the benjamin ono equation in l 2 mathbb t the lowest regularity where the initial value problem is well posed this problem was already initiated in linares and rosier trans am math soc 367 4595 4626 2015 where a stronger stabilization term was used that makes the equation of parabolic type in the control zone here we employ a more natural stabilization term related to the l2 norm moreover by proving a theorem of controllability in l2 we manage to prove the global controllability in large time our analysis relies strongly on the bilinear estimates proved in molinet and pilod anal pde 5 365 395 2012 and some new extension of these estimates established here

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

porosity of free boundaries in a obstacle problems abstract we establish the exact growth of the solution of the a obstacle problem near the free boundary from which we deduce its porosity

on global existence for the schrodinger ib system abstract in this paper we study global well posedness of the schrodinger improved boussinesq s ib system of equations in sobolev spaces we prove that the one and two dimensional s ib systems are globally well posed in l 2 r h 1 2 r h 1 2 r and l 2 r 2 l 2 r 2 l 2 r 2 h 1 r 2 respectively improving the corresponding results of ozawa and tsutaya t ozawa k tsutaya on the cauchy problem for schrodinger improved boussinesq equations adv stud pure math in press

on equations of kp type we discuss the local cauchy problem for the generalised kadomtsev petviashvili equation namely in the periodic and nonperiodic settings

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

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

blowup result for a type of generalized zakharov system abstract we study the blowup for a type of generalized zakharov system in this paper it is proved that the solution of such system either blows up in finite time or blows up in infinite time provided that the initial energy is negative

wellposedness for semirelativistic schrodinger equation with power type nonlinearity abstract in this paper we employ bourgain type norm to investigate the wellposedness for the cauchy problem of one dimensional semirelativistic schrodinger equation with cubic nonlinearity in space h s we extend the previous existence result to the space regularity s 1 4 moreover we point out the persistence of the solution in different regularity space is uniform by an iterate argument additionally we are also able to say more for semirelativistic schrodinger equation with general power type nonlinearity

on low regularity of the ostrovsky stepanyams and tsimring equation abstract in this paper we prove by using the fourier restriction norm method that the initial value problem of the ostrovsky stepanyams and tsimring 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 for any s 5 4 which improve our former result in zhao and cui 2009 5

on the porosity of free boundaries in degenerate variational inequalities abstract in this note we consider a certain degenerate variational problem with constraint identically zero the exact growth of the solution near the free boundary is established a consequence of this is that the free boundary is porous and therefore its hausdorff dimension is less than n and hence it is of lebesgue measure zero

generalized projection pursuit regression projection pursuit regression ppr can be used to estimate a smooth function of several variables from noisy and scattered data the estimate is a sum of smoothed one dimensional projections of the variables this paper discusses an extension of ppr to exponential family distributions called generalized projection pursuit regression gppr the proposed model allows multiple responses and nonlinear projections of the variables estimators are defined as minimizers of penalized cost functionals and estimation is related to the local scoring procedure for generalized additive models gams smooths are updated using a blockwise gauss seidel bgs method and convergence is shown for this procedure the use of generalized cross validation gcv to estimate smoothing parameters is discussed experimental results are shown for two types of data

on the cauchy problem for the generalized shallow water wave equation abstract in this paper we mainly study the cauchy problem for the generalized shallow water wave equation in the sobolev space h s of lower order s using the crucial bilinear estimates in the fourier transform restriction spaces related to the shallow water wave equation we establish local well posedness in h s with any s 11 16

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

supersonic limit for the zakharov rubenchik system abstract we study the asymptotic behavior of the solutions of zakharov rubenchik system when appropriate parameters go to zero namely we state weak and strong convergence results of these solutions to solutions of zakharov system the proof of the weak limit is a classical argument in the theory of compactness whose main ingredient is the aubin lions theorem and the ascoli theorem strong limits are conveniently treated by decomposing the nonlinearities and using the strichartz estimates associated with the group of the schrodinger equation and the wave group

automatic smoothing of regression functions in generalized linear models abstract we consider the penalized likelihood method for estimating nonparametric regression functions in generalized linear models nelder and wedderburn 1972 and present a generalized cross validation procedure for empirically assessing an appropriate amount of smoothing in these estimates asymptotic arguments and numerical simulations are used to show that the generalized cross validatory procedure preforms well from the point of view of a weighted mean squared error criterion the methodology adds to the battery of graphical tools for model building and checking within the generalized linear model framework included are two examples motivated by medical and horticultural applications

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

on the low regularity of the benney lin equation abstract we consider the low regularity of the benney lin equation u t u u x u x x x u x x u x x x x u x x x x x 0 we established the global well posedness for the initial value problem of benney lin equation in the sobolev spaces h s r for 0 s 2 improving the well posedness result of biagioni and linares h a biaginoi f linares on the benney lin and kawahara equation j math anal appl 211 1997 131 152 for s 2 we also prove some ill posedness issues

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

well posedness for the kadomtsev petviashvili ii equation and generalisations we show the local in time well posedness of the cauchy problem for the kadomtsev petviashvili ii equation for initial data in the non isotropic sobolev space h s 1 s 2 r 2 with s 1 j and s 2 0 on the h s 1 0 r 2 scale this result includes the full subcritical range without any additional low frequency assumption on the initial data more generally we prove the local in time well posedness of the cauchy problem for the following generalisation of the kp ii equation u t d x u x u 2 x x u yy 0 u 0 u 0 for 4 3 max 1 3 4 1 4 3 8 s 2 0 and u 0 h s 1 s 2 r 2 we deduce global well posedness for s 1 0 s 2 0 and real valued initial data

the electrostatic limit for the 3d zakharov system abstract we consider the vectorial zakharov system describing langmuir waves in a weakly magnetized plasma in its original derivation zakharov 1972 the evolution for the electric field envelope is governed by a schrodinger type equation with a singular parameter which is usually large in physical applications motivated by this we study the rigorous limit as this parameter goes to infinity by using some strichartz type estimates to control separately the fast and slow dynamics in the problem we show that the evolution of the electric field envelope is asymptotically constrained onto the space of irrotational vector fields

global well posedness in l 2 for the periodic benjamin ono equation we prove that the benjamin ono equation is globally well posed in h s bbb t for s ge 0 moreover we show that the associated flow map is lipschitz on every bounded set of h s 0 bbb t s ge 0 and even real analytic in this space for small times this result is sharp in the sense that the flow map if it can be defined and coincides with the standard flow map on h 0 infty bbb t cannot be of class c 1 alpha alpha 0 from h 0 s bbb t into h 0 s bbb t as soon as s 0

global well posedness of the kp i initial value problem in the energy space we prove that the kp i initial value problem begin cases partial tu partial x 3u partial x 1 partial y 2u partial x u 2 2 0 text on mathbb r 2 x y times mathbb r t u 0 phi end cases is globally well posed in the energy space mathbf e 1 mathbb r 2 big phi mathbb r 2 to mathbb r phi mathbf e 1 mathbb r 2 approx phi l 2 partial x phi l 2 big partial x 1 partial y phi big l 2 infty big

two asymptotic problems for a singular nonlinear schrodinger system in this paper we continue our investigation of the relation between various systems that can be derived from the klein gordon zakharov system in the high frequency and subsonic limits in this paper we start from the singular nonlinear schr odinger system which was derived in a previous work and derive the classical nonlinear schr odinger system in two different limit cases these two nonlinear schr odinger systems have different coefficients in the nonlinearity

space propagation of instabilities in zakharov equations in this paper we study an initial boundary value problem for the zakharov equations describing the space propagation of a laser beam entering a plasma we prove a strong instability result and prove that the mathematical problem is ill posed in sobolev spaces we also show that it is well posed in spaces of analytic functions several consequences for the physical consistency of the model are discussed

on the propagation of regularities in solutions of the fifth order kadomtsev petviashvili ii equation abstract in this paper we study special properties of solutions of the initial value problem ivp associated to the fifth order kadomtsev petviashvili ii equation mainly for initial data h 1 r 2 whose restriction belongs to h m x 0 r for some m z m 2 and x 0 r we prove that the restriction of the corresponding solution u t belongs to h m b r for any b r and any t 0 in other words the extra regularity on the data propagates in the solutions in the direction of the dispersion

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

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

on the convergence of the solution for a generalized zakharov system in this paper we study the limit behavior of the smooth solution for the generalized zakharov system with a magnetic field which describes the spontaneous generation of a magnetic field in a cold plasma by showing uniform bounds of solutions for this system with suitable initial data we prove that solution of the generalized zakharov system converges to the solution of the nonlinear schrodinger equation with a magnetic field

on a perturbation of the benjamin ono equation abstract we prove using the fourier restriction norm method that the initial value problem associated to a perturbation of the benjamin ono equation u t u u x h u x x h u x u x x 0 where x r t 0 0 and h denotes the usual hilbert transform is locally well posed in the sobolev space h s r for any s 1 2 and globally well posed in h s r when s 0 for s 1 we also prove some ill posedness issues

sharp bilinear estimates and its application to a system of quadratic derivative nonlinear schrodinger equations abstract in the present paper we consider the cauchy problem of the system of quadratic derivative nonlinear schrodinger equations for the spatial dimension d 2 and 3 this system was introduced by colin and colin 2004 the first author obtained some well posedness results in the sobolev space h s but under some condition for the coefficient of laplacian this result is not optimal we improve the bilinear estimate by using a nonlinear version of the classical loomis whitney inequality and prove well posedness in h s for s 1 2 if d 2 and s 1 2 if d 3

the well posedness of the dissipative korteweg de vries equations with low regularity data abstract we study the cauchy problem of a dissipative version of the kdv equation with rough initial data by working in a bourgain type space we prove the local and global well posedness results for sobolev spaces of negative order and the order number is lower than the well known value 3 4 in some sense this paper is intended to show how the bourgain type space is applicable to the study of semilinear equations with a linear part which contain both dissipative and dispersive terms

global well posedness for the kp i equation we prove the global well posedness of the kp i equation in bbb r 2 for data of arbitrary size in a suitable sobolev class we use a compactness method to get local solutions we extend the solutions thanks to some conservation laws an almost conserved quantity and strichartz estimates

smoothness of solutions of nonlinear ode smoothness of a c infty function f is measured by carleman sequence m m k 0 infty we say f in c infty m 0 1 if f k t leq cr k m k with c r 0 as an extension of our results in 2 we prove the following type statements let u be c infty function on 0 1 such that u prime phi u t where phi z t is an entire function of order 1 finite type on z and of a nonanalytic carleman class on t then u is in the same carleman class

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

on the nonlinear schrodinger limit of the magnetic zakharov system abstract in this paper we study the limit behavior of a smooth solution for the magnetic type zakharov system as the parameters tend to infinity we prove that solutions of the magnetic zakharov system converge to the solution of the nonlinear schrodinger equation moreover the detailed convergence rate is also investigated

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

comparison of numerical methods for the zakharov system in the subsonic limit regime abstract we compare numerically spatial temporal resolution of various methods for solving the zakharov system zs in the subsonic limit regime which involves a small parameter 0 e 1 inversely proportional to the acoustic speed in this regime i e 0 e 1 the solution presents highly oscillatory initial layers due to the wave operator or the incompatibility of the initial data specifically the solution propagates waves with wavelength of o e and o 1 in time and space respectively by applying the sine pseudospectral discretization for spatial derivatives followed by a time splitting technique for integrating the schrodinger equation combined with an exponential wave integrator in phase space for integrating the wave equation we propose four different numerical methods for the zs based on different quadrature rules for approximating the integral or some property of conservation numerical results suggest that all the methods are spectrally accurate in space which is uniformly for e 0 1 for temporal error the best method converges uniformly with linear convergence rate at o in the subsonic limit regime

variational inequalities in orlicz sobolev spaces on etudie l existence de solutions d inegalites variationnelles associees a des operateurs differentiels elliptiques

optimal lower bound estimates for the blow up rate for the zakharov system in a nonhomogeneous medium abstract in this paper we obtain lower bound estimates for the blow up rate of finite time blow up solutions to the cauchy problem for the zakharov system in a nonhomogeneous medium in two space dimensions by introducing suitable scale transformations of space and time and the use of compactness arguments we derive an optimal lower bound estimate in the energy space h 2 r 2 l 2 r 2 h 1 r 2 for the blow up rate for t near the finite blow up time t also we give an application to the virial identity for the zakharov system under study

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

on finite energy solutions of the kp i equation we prove that the flow map of the kadomtsev petviashvili i kp i equation is not uniformly continuous on bounded sets of the natural energy space

blow up and nonlinear instability for the magnetic zakharov system abstract this study deals with the generalized zakharov system with magnetic field first of all we construct a kind of blow up solutions and establish the existence of blow up solutions to the system through considering an elliptic system next we show the nonlinear instability for a kind of periodic solutions in addition we consider the concentration properties of blow up solutions for the system under study at the end of this paper we establish the global existence of weak solutions to the cauchy problem of the system under consideration

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 the low regularity of the fifth order kadomtsev petviashvili i equation we consider the fifth order kadomtsev petviashvili i kp i equation as tu 3 x u 5 x u 1 x 2 y u uux 0 while r we introduce an interpolated energy space es to consider the well posedeness of the initial value problem ivp of the fifth order kp i equation we obtain the local well posedness of ivp of the fifth order kp i equation in es for 0 s 1 to obtain the local well posedness we present a bilinear estimate in the bourgain space in the framework of the interpolated energy space it crucially depends on the dyadic decomposed strichartz estimate the fifth order dispersive smoothing effect and maximal estimate

local well posedness for the modified kdv equation in almost critical widehat h r_s spaces we study the cauchy problem for the modified kdv equation u t u xxx u 3 x 0 u 0 u 0 for data u 0 in the space h r s defined by the norm u 0 h r s s u 0 lr local well posedness of this problem is established in the parameter range 2 r 1 s 1 2 1 2r so the case s r 0 1 which is critical in view of scaling considerations is almost reached to show this result we use an appropriate variant of the fourier restriction norm method as well as bi and trilinear estimates for solutions of the airy equation

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

time dependent effects of fixed covariates in cox regression a nonparametric modification is proposed for cox s proportional hazards model cox 1972 journal of the royal statistical society series b 34 187 220 where the covariates are fixed but their effects are allowed to vary in time parameters are introduced for the covariate effects at the uncensored survival times estimates are obtained by maximizing the penalized partial log likelihood that arises if a penalty function of the first order differences of consecutive parameters is subtracted from the partial log likelihood the choice of the smoothing parameter which determines the weight of the penalty is based on akaike s information criterion the methods are illustrated in a set of ovarian cancer data and in a set of kidney transplantation data

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

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

survival analysis and neural nets we consider feed forward neural nets and their relation to regression models for survival data we show how the back propagation algorithm may be used to obtain maximum likelihood estimates in certain standard regression models for survival data as well as in various generalizations of these examples concerning malignant melanoma and post partum amenorrhoea during lactation are used as illustration we conclude that although problems with the substantial number of parameters and their interpretation remain the feed forward neural network models are flexible extensions to the standard regression models and thereby candidates for use in prediction and exploratory analyses in larger data sets

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

well posedness for the cauchy problem for a system of semirelativistic equations the local well posedness for the cauchy problem of a system of semirelativistic equations in one space dimension is shown in the sobolev space hs of order s 0 we apply the standard contraction mapping theorem by using bourgain type spaces xs b we also use an auxiliary space for the solution in l2 h0 we give the global well posedness by this conservation law and the argument of the persistence of regularity

quasi invariant gaussian measures for the cubic fourth order nonlinear schrodinger equation we consider the cubic fourth order nonlinear schrodinger equation on the circle in particular we prove that the mean zero gaussian measures on sobolev spaces h s mathbb t s frac 3 4 are quasi invariant under the flow

well posedness of the cauchy problem of ostrovsky equation in anisotropic sobolev spaces abstract we study the cauchy problem of the ostrovsky equation t u x 3 u x 1 u u x u 0 with 0 by establishing a bilinear estimate on the anisotropic bourgain space x s b we prove that the cauchy problem of this equation is locally well posed in the anisotropic sobolev space h s r for any s 5 8 and some 0 1 2 using this result and conservation laws of this equation we also prove that the cauchy problem of this equation is globally well posed in h s r for s 0

derivation of the zakharov equations this article continues the study initiated in 27 7 of the validity of the zakharov model which describes langmuir turbulence we give an existence theorem for a class of singular quasilinear equations this theorem is valid for prepared initial data we apply this result to the euler maxwell equations which describes laser plasma interactions to obtain in a high frequency limit an asymptotic estimate that describes solutions of the euler maxwell equations in terms of wkb approximate solutions the leading terms of which are solutions of the zakharov equations due to the transparency properties of the euler maxwell equations evidenced in 27 this study is carried out in a supercritical highly nonlinear regime in such a regime resonances between plasma waves electromagnetric waves and acoustic waves could create instabilities in small time the key of this work is the control of these resonances the proof involves the techniques of geometric optics of joly metivier and rauch 12 13 recent results by lannes on norms of pseudodifferential operators 14 and a semiclassical paradifferential calculus

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

the cauchy problem for higher order kp equations abstract we study the local well posedness of higher order kp equations our well posedness results make an essential use of a global smoothing effect for the linearized equation established in ben artzi and saut preprint 1997 injected into the framework of fourier transform restriction spaces introduced by bourgain our ill posedness results rely on the existence of solitary wave solutions and on scaling arguments the method was first applied in the context of the kdv and schrodinger equations cf birnir et al ann inst h poincare anal non lineaire 13 1996 529 535 birnir et al j london math soc 53 1996 551 559

sharp ill posedness result for the periodic benjamin ono equation abstract we prove the discontinuity for the weak l 2 t topology of the flow map associated with the periodic benjamin ono equation this ensures that this equation is ill posed in h s t as soon as s 0 and thus completes exactly the well posedness result obtained in molinet 2008 12

cauchy problem of schrodinger improved boussinesq systems on the torus in this paper we will study the local well posedness of schrodinger improved boussinesq system with additive noise in td d 1 and we will also study the global well posedness of dimension 1 case with the initial data u0 v1 v2 l2 l2 l2 almost surely gaining some exponential growth of l2 norm of v

on the low regularity of the modified korteweg de vries equation with a dissipative term abstract we consider a dissipative version of the modified korteweg de vries equation u t u x x x u x x u 3 x 0 we prove global well posedness results on the associated cauchy problem in the sobolev spaces h s r for s 1 4 while for s 1 2 we prove some ill posedness issues

1 soliton solution of the generalized kp equation with generalized evolution this paper obtains the 1 soliton solution of three variants of the generalized kp equation with generalized evolution the solitary wave ansatz is used to carry out the integration of such equations the parameter domain is also identified in the process the numerical simulations are also obtained in this paper

on the existence and uniqueness of smooth solution for a generalized zakharov equation abstract the paper deals with the existence and uniqueness of smooth solution for a generalized zakharov equation we establish local in time existence and uniqueness in the case of dimension d 2 3 moreover by using the conservation laws and brezis gallouet inequality the solution can be extended globally in time in two dimensional case for small initial data besides we also prove global existence of smooth solution in one spatial dimension without any small assumption for initial data

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

well posedness and weak rotation limit for the ostrovsky equation we consider the cauchy problem of the ostrovsky equation we first prove the time local well posedness in the anisotropic sobolev space hs a with s a 2 3 4 and 0 a 1 by the fourier restriction norm method this result include the time local well posedness in hs with s 3 4 for both positive and negative dissipation namely for both 0 and 0 we next consider the weak rotation limit we prove that the solution of the ostrovsky equation converges to the solution of the kdv equation when the rotation parameter goes to 0 and the initial data of the kdv equation is in l2 to show this result we prove a bilinear estimate which is uniform with respect to

gevrey regularity for a class of water wave models abstract local well posedness for a class of higher order nonlinear dispersive partial differential equations is obtained in spaces of functions analytic on a strip around the real axis the proof relies on estimates in space time norms adapted to the linear part of the equation the class of equations in view contains a number of equations arising in the modeling of waves in fluids and also in other applications