on the structural stability of the euler voigt and navier stokes voigt models abstract we consider the euler voigt equations and the navier stokes voigt equations which are obtained by an inviscid regularization from the corresponding equations the main result we show is the structural stability of the system in terms of the variations of both viscosity and regularization parameters

well posedness of korteweg de vries benjamin bona mahony equation on a finite domain abstract in this paper we consider the korteweg de vries benjamin bona mahony equation on a finite domain with initial value and nonhomogeneous boundary conditions this particular problem arises from the phenomenon of long wave with small amplitude in fluid we get the global well posedness of this system

global solutions to the two dimensional quasi geostrophic equation with critical or super critical dissipation the two dimensional quasi geostrophic 2d qg equation with critical and super critical dissipation is studied in sobolev space h s 2 for critical case open image in new window existence of global large solutions in h s is proved for s open image in new window when open image in new window is small this generalizes and improves the results of constantin d cordoba and wu 4 for s 1 2 and the result of a cordoba and d cordoba 8 for s open image in new window for s 1 these solutions are also unique the improvement for pushing s down from 1 to open image in new window is somewhat surprising and unexpected for super critical case 0 open image in new window existence and uniqueness of global large solution in h s is proved when the product open image in new window is small for suitable s 2 2 p 1 and 0 1

global regularity for a class of 2d generalized tropical climate models abstract this paper establishes the global existence and regularity of solutions to a two dimensional 2d tropical climate model tcm with fractional dissipation the inviscid counterpart of this model was derived by frierson majda and pauluis 8 as a model for tropical geophysical flows this model reflects the interaction and coupling among the barotropic mode u the first baroclinic mode v of the velocity and the temperature the systems with fractional dissipation studied here may arise in the modeling of geophysical circumstances mathematically these systems allow simultaneous examination of a family of systems with various levels of regularization the aim here is the global regularity with the least dissipation we prove two main results first the global regularity of the system with v and for 1 and 3 2 and second the global regularity of the system with v for 3 2 the proofs of these results are not trivial and the requirements on the fractional indices appear to be optimal the key tools employed here include the maximal regularity for general fractional heat operators the littlewood paley decomposition and besov space techniques lower bounds involving fractional laplacian and simultaneous estimates of several coupled quantities

computing the solution of the korteweg de vries equation with arbitrary precision on turing machines in this paper we answer an open question raised by pour el and richards is the solution operator of the korteweg de vries kdv equation computable the initial value problem of the kdv equation posed on the real line r ut uux uxxx o t x r u x 0 x defines a nonlinear map kr from the space hs r to the space c r hs r for real numbers s 0 we prove that for any integer s 3 the map kr hs r c r hs r is turing computable

global well posedness for a boussinesq navier stokes system with critical dissipation abstract in this paper we study a fractional diffusion boussinesq model which couples a navier stokes type equation with fractional diffusion for the velocity and a transport equation for the temperature we establish global well posedness results with rough initial data

a higher order triangular plate bending element revisited a new formulation of an eighteen degrees of freedom higher order triangular plate bending element using triangular area co ordinates is presented the displacement function w is taken as the complete fifth order polynomial in area co ordinates the normal slope along an edge of the triangle is constrained to vary cubically the twenty one constants are expressed explicity in terms of eighteen degrees of freedom the element stiffness matrix is expressed as a product of component matrices for which explicit expressions are developed and presented no numerical inversion or integration is necessary the formulation is expected to be useful specially for microcomputers

coupled error spending functions for parallel bivariate sequential tests sequential procedures are developed to facilitate marginal monitoring of bivariate response vectors in clinical trials the general approach is based on an extension of the error spending function methodology of lan and demets 1983 biometrika 70 659 663 and is sufficiently flexible that one may elect to fix the experimental type i error rate cook 1994 controlled clinical trials 15 3 187 200 or the marginal type i error rates sample size calculations are described to ensure power requirements are satisfied for marginal tests of significance reformulating the procedures in terms of repeated confidence intervals jennison and turnbull 1989 journal of the royal statistical society series b 51 305 361 lends added flexibility to the monitoring process the developments are discussed in the context of responses with a bivariate normal distribution data from an asthma intervention trial are used for illustrative purposes

local well posedness for the maxwell schrodinger equation time local well posedness for the maxwell schrodinger equation in the coulomb gauge is studied in sobolev spaces by the contraction mapping principle the lorentz gauge and the temporal gauge cases are also treated by the gauge transform

global regularity for the 2d mhd equations with mixed partial dissipation and magnetic diffusion abstract whether or not classical solutions of the 2d incompressible mhd equations without full dissipation and magnetic diffusion can develop finite time singularities is a difficult issue a major result of this paper establishes the global regularity of classical solutions for the mhd equations with mixed partial dissipation and magnetic diffusion in addition the global existence conditional regularity and uniqueness of a weak solution is obtained for the 2d mhd equations with only magnetic diffusion

a strong type 2 2 estimate for a maximal operator associated to the schrodinger equation soit t f x sup t 0 t t f x ou t t f 1 exp it 2 f 1 4 on montre que etant donne un intervalle fini i i t f 2 dx c i r f x 2 dx

sizing clinical trials when comparing bivariate time to event outcomes clinical trials with multiple primary time to event outcomes are common use of multiple endpoints creates challenges in the evaluation of power and the calculation of sample size during trial design particularly for time to event outcomes we present methods for calculating the power and sample size for randomized superiority clinical trials with two correlated time to event outcomes we do this for independent and dependent censoring for three censoring scenarios i the two events are non fatal ii one event is fatal semi competing risk and iii both are fatal competing risk we derive the bivariate log rank test in all three censoring scenarios and investigate the behavior of power and the required sample sizes separate evaluations are conducted for two inferential goals evaluation of whether the test intervention is superior to the control on 1 all of the endpoints multiple co primary or 2 at least one endpoint multiple primary copyright 2017 john wiley sons ltd

group sequential enrichment design incorporating subgroup selection an important component of clinical trials in drug development is the analysis of treatment efficacy in patient subgroups subpopulations because of concerns of multiplicity and of the small sample sizes often involved such analyses can present substantial statistical challenges and may lead to misleading conclusions as a confirmatory seamless phase ii iii design we propose an adaptive enrichment group sequential procedure whereby resources can be concentrated on subgroups most likely to respond to treatment stopping boundaries are defined through upper and lower spending functions the procedure is presented in terms of the efficient score enabling the analysis of normal binary or time to event data it addresses the dilution effect by eliminating populations at the first stage that appear likely to derive no therapeutic benefit it subsequently proceeds with the definitive assessment of treatment efficacy among the remaining pooled populations using a group sequential design the procedure provides strong protection of familywise type i error rate and we employ a bootstrap algorithm to obtain point and interval estimates that are adjusted for the selection bias we give examples to demonstrate how the design is used we make comparisons with adaptive two stage combination test procedures and with a group sequential test that does not account for the presence of subgroups numerical results show that the procedure has high power to detect subgroup specific effects and the use of multiple interim analysis points can lead to substantial sample size savings

nonlinear small data scattering for the wave and klein gordon equation on etudie l operateur diffusion qui appartient a la paire d equations u tt au f u 0 et u tt au 0 a represente l operateur j 1 n 2 x j 2 m 2 m r et f u u 1 u ou r 1

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous galerkin formulation abstract when considering problems of dimensions close to the characteristic length of the material the size effects can not be neglected and the classical so called first order multiscale computational homogenization scheme fmch looses accuracy motivating the use of a second order multiscale computational homogenization smch scheme this second order scheme uses the classical continuum at the micro scale while considering a second order continuum at the macro scale although the theoretical background of the second order continuum is increasing the implementation into a finite element code is not straightforward because of the lack of high order continuity of the shape functions in this work we propose a smch scheme relying on the discontinuous galerkin dg method at the macro scale which simplifies the implementation of the method indeed the dg method is a generalization of weak formulations allowing for inter element discontinuities either at the c 0 level or at the c 1 level and it can thus be used to constrain weakly the c 1 continuity at the macro scale the c 0 continuity can be either weakly constrained by using the dg method or strongly constrained by using usual c 0 displacement based finite elements therefore two formulations can be used at the macro scale i the full discontinuous galerkin formulation fdg with weak c 0 and c 1 continuity enforcements and ii the enriched discontinuous galerkin formulation edg with high order term enrichment into the conventional c 0 finite element framework the micro problem is formulated in terms of standard equilibrium and periodic boundary conditions a parallel implementation in three dimensions for non linear finite deformation problems is developed showing that the proposed method can be integrated into conventional finite element codes in a straightforward and efficient way

on the stability of internal waves the extended kdv equation ut uux u2ux uxxx 0 is widely used as a model describing internal waves in ideal fluids the equation admits a family of negative and positive solitary waves c these solitary waves exhibit the typical broadening effect seen in internal waves it is shown here that all solitary wave solutions of the extended kdv equation are orbitally stable the proof of stability is based on the general theory of grillakis et al 1987 j funct anal 74 160 for equations of the form ut je u which have two conserved integrals e u and v u a spectral analysis of the linear operator reduces the question of orbital stability to the question of whether the scalar function d c e c cv c is convex to prove the stability an explicit calculation showing the convexity is performed

implicit gradient elasticity a new formulation of gradient elasticity is derived and implemented pade approximations are used to introduce an implicit relation between non local strains and displacements as a result the finite element interpolation requires only 0 continuous rather than 1 continuous shape functions the underlying energy functional is presented and it is found that the present formulation is of the mixed type whereby the non local strains act as the primary unknowns and the displacements as the auxiliary unknowns the implications on the interpolation are studied finally the influence of the additional length scale parameter on the global response is assessed copyright 2006 john wiley sons ltd

global well posedness bkm blow up criteria and zero h limit for the 3d incompressible hall mhd equations abstract we consider the cauchy problem for the 3d incompressible hall mhd equations firstly we prove the global well posedness under some new conditions which improve the previous corresponding results in 3 4 24 then a beale kato majda bkm blow up criterion is established finally we show the explicit convergence rate of strong solutions from the hall mhd equations to the mhd equations as the hall coefficient h 0

analysis of time fractional and space nonlocal stochastic nonlinear schrodinger equation driven by multiplicative white noise abstract in this paper we establish the existence and uniqueness of the mild solution and the holder continuity for the time fractional and space nonlocal stochastic nonlinear schrodinger equation driven by multiplicative white noise a full discrete scheme with spectral galerkin method in space and exponential method in time is proposed it is rigorously proved that the numerical scheme admits an unique solution finally we present the optimal error estimates of the proposed scheme based on the regularity analysis

stability of solitary waves for a nonlinear dispersive system in a critical case we study a nonlinear dispersive system with critical power nonlinearities building on the works of weinstein and angulo bona linares and scialom a conditional stability modulo the group of translations in space and dilations is proved for a solitary wave solution

sharp local well posedness for a fifth order shallow water wave equation abstract in this paper we prove that the already established local well posedness in the range s 5 4 of the cauchy problem with an initial h s r data for a fifth order shallow water wave equation is extendable to s 5 4 by using the f s space this is sharp in the sense that the ill posedness in the range s 5 4 of this initial value problem is already known

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

a higher dispersion kdv equation on the line abstract the cauchy problem for a korteweg devries equation with dispersion of order m 2 j 1 where j is a positive integer kdvm is studied with data in sobolev and analytic spaces first optimal bilinear estimates in bourgain spaces are proved and using them well posedness in sobolev spaces h s s j 1 4 is established then well posedness in analytic gevrey spaces g s 0 is proved by using an analytic version of the bilinear estimates this implies that the uniform radius of analyticity persist for some time for the later times a lower bound for the radius of spacial analyticity is derived which is given by t c t with 4 3 e for any e 0 when j 1 and 1 when j 2 finally it is shown that the regularity of the solution in the time variable is gevrey of order m and this is optimal

local well posedness for the nonlocal derivative nonlinear schrodinger equation in besov spaces abstract in this paper we study the cauchy problem associated with the one dimensional integro differential nonlocal derivative nonlinear schrodinger equation in the besov space b 2 1 2 1 r the local well posedness for small initial data in b 2 1 2 1 r is established our method of proof combines the contraction principle applied to the associated integral equation together with interpolations of some smoothing effects kato s smoothing effects strichartz estimate and estimates for the maximal function for phase localized functions associated to the linear dispersive part of the equation and a fractional vector valued leibniz s rule derived by molinet and ribaud in 2004

nonlinear small data scattering for the generalized korteweg de vries equation abstract we study the longtime stability of small solutions to the ivp for the generalized korteweg de vries equation we obtain a lower bound for the degrees of nonlinearity of the perturbation which guarantees that the small solutions of the nonlinear problem behave asymptotically like the solutions of the associated linear problem this behavior allows us to establish a nonlinear scattering result for small perturbations with these degrees for a class of small data we improve the value of this lower bound the new crucial ingrediants in our proofs are the l p decay estimates of the half derivative of the airy kernel

local existence for the non resistive mhd equations in besov spaces in this paper we prove the existence of solutions to the viscous non resistive magnetohydrodynamics mhd equations on the whole of rn n 2 3 for divergence free initial data in certain besov spaces namely u0 bn 2 1 2 1 and b0 bn 2 2 1 the a priori estimates include the term t 0 u s 2 hn 2 ds on the right hand side which thus requires an auxiliary bound in hn 2 1 in 2d this is simply achieved using the standard energy inequality but in 3d an auxiliary estimate in h1 2 is required which we prove using the splitting method of calderon 1990 2 by contrast our proof that such solutions are unique only applies to the 3d case

key multiplicity issues in clinical drug development much progress has been made over the past decade with the development of novel methods for addressing increasingly more complex multiplicity problems arising in confirmatory phase iii clinical trials this includes traditional problems with a single source of multiplicity for example analysis of multiple endpoints or dose placebo contrasts in addition more advanced problems with several sources of multiplicity have attracted attention in clinical drug development these problems include two or more families of objectives such as multiple endpoints evaluated at multiple dose levels or in multiple patient populations this paper provides a review of concepts that play a central role in defining and solving multiplicity problems error rate definitions and introduces main classes of multiple testing procedures widely used in clinical trials nonparametric semiparametric and parametric procedures the paper also presents recent advances in multiplicity research including gatekeeping procedures for clinical trials with multiple sets of objectives the concepts and methods introduced in the paper are illustrated using several case studies on the basis of real clinical trials software implementation of commonly used multiple testing and gatekeeping procedures is discussed copyright 2012 john wiley sons ltd

global small solutions to a tropical climate model without thermal diffusion we obtain the global well posedness of classical solutions to a tropical climate model with only the dissipation of the first baroclinic model of the velocity v under small initial data this model is a modified version of the original system derived by frierson majda pauluis in frierson et al commun math sci 2 591 626 2004 the main difficulty is the absence of thermal diffusion to overcome it we exploit the structure of the equations coming from the coupled terms dissipation term and damp term then we find the hidden thermal diffusion in addition based on the littlewood paley theory we establish a generalized commutator estimate which may be applied to other partial differential equations

challenge of multiple co primary endpoints a new approach there are many disorders where regulatory agencies have required a new treatment to demonstrate efficacy on multiple co primary endpoints all significant at the one sided 2 5 per cent level before accepting the treatment s effect for the disorder this requirement rooted in the intersection union iu test has led many researchers to increase the study sample size to make up for the reduction in the statistical power at the study level unfortunately the increase in sample size could be substantial when the endpoints are minimally correlated and the treatment effects on the multiple endpoints are comparable in this paper we demonstrate that the frequentist concept of controlling the maximum false positive rate even when applied to a restricted null space has only limited success in keeping the sample size increase at a reasonable level we therefore propose an approach that is based on the notion of controlling an average type i error rate by employing an upper bound for the average type i error rate the new approach provides an adjustment to the significance level that depends only on the correlation among the endpoints for the most common case of two or three co primary endpoints the adjusted significance level is at most 5 per cent one sided when the endpoints are moderately correlated we show how sample size could be calculated under the proposed approach and contrast the needed sample size with that required under the iu test we provide additional comments and discuss why the new approach is consistent with the principle requiring evidence of significance in the drug development and approval process

discrete sequential boundaries for clinical trials summary pocock 1977 o brien fleming 1979 and slud wei 1982 have proposed different methods to construct discrete sequential boundaries for clinical trials these methods require that the total number of decision times be specified in advance in the present paper we propose a more flexible way to construct discrete sequential boundaries the method is based on the choice of a function a t which characterizes the rate at which the error level ac is spent the boundary at a decision time is determined by a t and by past and current decision times but does not depend on the future decision times or the total number of decision times

asymptotic expansion for solutions of the navier stokes equations with potential forces abstract we derive an asymptotic expansion for smooth solutions of the navier stokes equations in weighted spaces this result removes previous restrictions on the number of terms of the asymptotics as well as on the range of the polynomial weights we also write the expansion in terms of expressions involving non linear quantities

two new triangular g1 conforming finite elements with cubic edge rotation for the analysis of kirchhoff plates abstract two triangular g 1 conforming elements based on the triangular gregory patch suitable for the analysis of the kirchhoff plate model are presented both have cubic normal derivative along the sides so that they can be effectively used in combination with generalized hermitian elements the gregory patch consists in a rational enhancement of the base polynomial spaces useful to design g 1 conforming elements on general c 0 conforming unstructured meshes because of the presence of the rational functions the second derivatives at the corners of the element present finite discontinuities that prevent the elements from passing the bending patch test the discontinuities are removed using a constrained version of the gregory patch with lagrange multipliers in this way the rational conforming space collapses into a conforming rearrangement of the original polynomial interpolant spaces the proposed formulation design elements that pass the bending patch test and present optimal rate of convergence on general unstructured meshes

a new c1 finite element full heptic a new c1 finite element complete in 7th degree polynomial basis is described and compared with other c1 elements in an eigen elements analysis

almost periodicity in time of solutions of the kdv equation we study the cauchy problem for the kdv equation tu 6u xu xu 0 with almost periodic initial data u x 0 v x we consider initial data v for which the associated schrodinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify and show the existence uniqueness and almost periodicity in time of solutions this establishes a conjecture of percy deift for this class of initial data the result is shown to apply to all small analytic quasiperiodic initial data with diophantine frequency vector

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

adaptive group sequential design with population enrichment in phase 3 randomized controlled trials with two binary co primary endpoints the use of co primary endpoints in drug development allows investigators to capture an experimental intervention s multidimensional effect more comprehensively than a single primary endpoint we propose the theoretical basis and development of an adaptive population enrichment design with co primary endpoints provide stage wise boundary values for futility and efficacy and discuss power under different efficacy configurations subgroup prevalence and analysis times using a pre specified decision criterion we considered a two arm two stage parallel group design where population enrichment occurs at the interim analysis by dropping any non responsive subgroups a test for efficacy is conducted only in the enriched population two binary endpoints are evaluated as co primary endpoints our trial objective is to determine whether the experimental intervention is superior to the control intervention with superiority required in both endpoints we define the stopping boundary using alpha spending functions using a 0 025 significance level for each endpoint we obtain the stage i threshold boundary values for futility and efficacy as 0 1040 and 2 2761 respectively and the stage ii boundary value for futility and efficacy is 2 2419 we show that in the presence of substantial heterogeneity of treatment effect we gain more power to observe an effect in the subgroup where the benefits are greater by allowing the dropping of non responsive subgroups at an early stage our design reduces the likelihood of obtaining false negative results due to inclusion of the heterogeneous treatment effects of both subgroups which would dilute the responsive subgroup s results

the fefferman stein decomposition for the constantin lax majda equation regularity criteria for inviscid fluid dynamics revisited the celebrated beale kato majda bkm criterion for the 3d euler equations has been updated by kozono and taniuchi by replacing the supremum with the bounded mean oscillation norm we consider this generalized criterion in an attempt to understand it more intuitively by giving an alternative explanation for simplicity we first treat the constantin lax majda clm equation t h for the vorticity in one dimension and identify a mechanism underlying the update of such an estimate we consider a fefferman stein fs decomposition for the initial vorticity 0 h 1 and how it propagates under the dynamics of the clm equation in particular we obtain a set of dynamical equations for it which reads in its simplest case 0 t 0h 0 1h 1 and 1 t 0h 1 1h 0 the equation for the second component 1 responsible for a possible logarithmic blow up is linear and homogeneous hence it remains zero if it is so initially until a stronger blow up takes place this rules out a logarithmic blo

long time dynamics of 2d quasi geostrophic equations with damping in lp abstract the long time behavior of solutions of 2d quasi geostrophic equations with damping on the whole space is studied the existence of the global attractor in l p is proved new difficulties come from nonlocal terms and the banach space l p we overcome them by using commutator estimates and positivity inequalities associated with fractional laplacian

method of balanced adjustment in testing co primary endpoints in a clinical trial if there are three or more co primary endpoints the type ii error could increase depending on the correlation among the endpoints and their treatment effect sizes to keep the type ii error under control one may have to consider larger sample sizes however in cases where treatment effect size of at least one of the endpoints is likely to be small the required sample size estimates can exceed reasonable bounds patel 1991 proposed an approach that adjusts the significance level for testing each primary endpoint based on the idea of restricting the null space in chuang stein et al 2007 the upward adjustment to the significance levels is based on controlling an average type i error rate in the scenario that statistical significance of each individual hypothesis is not required we introduce a compromise testing approach in which the significance level for a co primary endpoint is adjusted upward only if the treatment shows high significance in one or more than one of the remaining co primary endpoints the adjustment depends on the correlation among the endpoints larger adjustment is needed for cases of smaller correlation the method is applicable for the scenario where the null space is restricted our testing approach controls maximum joint false positive rate over the restricted null space

sample size determination in group sequential clinical trials with two co primary endpoints we discuss sample size determination in group sequential designs with two endpoints as co primary we derive the power and sample size within two decision making frameworks one is to claim the test intervention s benefit relative to control when superiority is achieved for the two endpoints at the same interim timepoint of the trial the other is when superiority is achieved for the two endpoints at any interim timepoint not necessarily simultaneously we evaluate the behaviors of sample size and power with varying design elements and provide a real example to illustrate the proposed sample size methods in addition we discuss sample size recalculation based on observed data and evaluate the impact on the power and type i error rate copyright 2014 john wiley sons ltd

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

global regularity for a class of 2d tropical climate model abstract in this paper we establish the global regularity of a modified version of the tropical climate model originally derived by frierson majda pauluis 2004 6 which couples the barotropic and baroclinic modes of the velocity and the typical midtropospheric temperature in proving the global regularity to bypass the difficulty caused by the very weak dissipation any small power 0 of the barotropic mode we utilize the weakly nonlinear energy estimate approach and the maximal l t q l x p regularity for the heat kernel

higher order commutator estimates and local existence for the non resistive mhd equations and related models this paper establishes the local in time existence and uniqueness of strong solutions in hs for s n 2 to the viscous non resistive magnetohydrodynamics mhd equations in rn n 2 3 as well as for a related model where the advection terms are removed from the velocity equation the uniform bounds required for proving existence are established by means of a new estimate which is a partial generalisation of the commutator estimate of kato ponce comm pure appl math 41 7 891 907 1988

local well posedness for the super korteweg de vries equation abstract we establish some qualitative properties for the cauchy problem associated with the super korteweg de vries equation super kdv super kdv t u x 3 u 1 2 x u 2 1 2 x 2 v 2 0 t v x 3 v x u v 0 x t r we prove local well posedness in weighted sobolev spaces x s 3 h s r h 3 x 2 d x h s r h 3 x 2 d x s 5 integer provided that the initial data is small enough and also in x s 11 h s r h 11 x 2 d x h s r h 11 x 2 d x with s 15 integer for arbitrary initial data the main ingredients of the proof for the first case are new estimates describing the smoothing effect of kato type for the kdv group w t t r that for the second case is via a change of variable performed and a deduction of new smoothing effects related to the kdv c kenig g staffilani local well posedness for higher order nonlinear dispersive systems j fourier anal appl 3 4 1997

regularity of solutions to the schrodinger equation soit f dans l espace de schwartz s r n et soit s t f x u x t rn exp ix exp it 2 f d x r n t r f est la transformee de fourier de f on considere u solution de l equation de schrodinger u i u t on pose s f x sup 0 t 1 s t f x x r n on introduit des espaces de sobolev h s en posant h s f s f hd s s r ou f hd s rn 1 2 s f 2 d 1 2 on etudie pour quelles valeurs de s on a b s f x 2 dx 1 2 c b f hd s f s r n pour toutes les boucles b de r n

a method for creating a class of triangular c1 finite elements finite elements providing a c1 continuous interpolation are useful in the numerical solution of problems where the underlying partial differential equation is of fourth order such as beam and plate bending and deformation of strain gradient dependent materials although a few c1 elements have been presented in the literature their development has largely been heuristic rather than the result of a rational design to a predetermined set of desirable element properties therefore a general procedure for developing c1 elements with particular desired properties is still lacking this paper presents a methodology by which c1 elements such as the tuba 3 element proposed by argyris et al can be constructed in this method which to the best of our knowledge is the first one of its kind a class of finite elements is first constructed by requiring a polynomial interpolation and prescribing the geometry the location of the nodes and the possible types of nodal dofs a set of necessary conditions is then imposed to obtain appropriate interpolations generic procedures are presented which determine whether a given potential member of the element class meets the necessary conditions the behaviour of the resulting elements is checked numerically using a benchmark problem in strain gradient elasticity

large strain finite element analysis of a local second gradient model application to localization summary a large strainnite element formulation based on a local second gradient plasticity model is presented the corresponding constitutive equations were developed as a direct extension of microstructured siam j appl math 1973 25 3 556 575 arch rational mech anal 1964 16 51 78 or micromorphic in mechanics of generalized continua iutam symposium kroner ed springer berlin 1968 18 35 j math mech 1966 15 6 909 924 continua in which a mathematical constraint between the micro kinematics description and the usual macrodeformation gradienteld has been introduced this constraint is enforced in a weak sense by the use of lagrange multipliers in order to avoid diculties with the c 1 continuity for thenite element method correspondingnite elements are then constructed involving the lagrange multiplierseld a geometrically non linear 2 dnite element code is developed within a framework of an incremental method for every step a full newton raphson involving a numerical consistent tangent stiness operator for the complete model i e the second gradient terms as well as the classical ones is done and some numerical tests allow to validate the method and to discuss the inuence of the geometrical non linearity copyright 2002 john wiley sons ltd

sample size determination in clinical trials with multiple co primary binary endpoints clinical trials often employ two or more primary efficacy endpoints one of the major problems in such trials is how to determine a sample size suitable for multiple co primary correlated endpoints we provide fundamental formulae for the calculation of power and sample size in order to achieve statistical significance for all the multiple primary endpoints given as binary variables on the basis of three association measures among primary endpoints we discuss five methods of power and sample size calculation the asymptotic normal method with and without continuity correction the arcsine method with and without continuity correction and fisher s exact method for all five methods the achieved sample size decreases as the value of association measure increases when the effect sizes among endpoints are approximately equal in particular a high positive association has a greater effect on the decrease in the sample size on the other hand such a relationship is not very strong when the effect sizes are different

large strain two scale computational approach using c1 continuity finite element employing a second gradient theory a two scale computational algorithm employing second order computational homogenization under assumption of large strain is presented a micro macro transition has been proposed where the c1 continuity displacement based finite element is used for discretization at the macrolevel while the representative volume element at the microlevel is discretized by c0 continuity finite elements the performed finite element verifications show an advantage in comparison to available finite element formulations in order to improve numerical efficiency of the proposed computational procedures the algorithms are embedded into the finite element program abaqus the performance of the presented multiscale homogenization approach is demonstrated in numerical examples where the elastoplastic deformation responses are considered and compared with the small strain analysis

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

energy conservation for the 3d tropical climate model in bounded domains abstract in this paper we consider the energy conservation to the 3d tropical climate model in bounded domain by exploring the special structure of the nonlinear components and using the ideas introduced in 1 we can give some sufficient conditions for the weak solutions to ensure energy conservation

cell based maximum entropy approximants in this paper we devise cell based maximum entropy max ent basis functions that are used in a galerkin method for the solution of partial differential equations the motivation behind this work is the construction of smooth approximants with controllable support on unstructured meshes in the variational scheme to obtain max ent basis functions the nodal prior weight function is constructed from an approximate distance function to a polygonal curve in r2 more precisely we take powers of the composition of r functions via boolean operations the basis functions so constructed are nonnegative smooth linearly complete and compactly supported in a neighbor ring of segments that enclose each node the smoothness is controlled by two positive integer parameters the normalization order of the approximation of the distance function and the power to which it is raised the properties and mathematical foundations of the new compactly supported approximants are described and its use to solve two dimensional elliptic boundary value problems poisson equation and linear elasticity is demonstrated the sound accuracy and the optimal rates of convergence of the method in sobolev norms are established

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

a three dimensional c1 finite element for gradient elasticity in gradient elasticity strain gradient terms appear in the expression of virtual work leading to the need for c1 continuous interpolation in finite element discretizations of the displacement field only employing such interpolation is generally avoided in favour of the alternative methods that interpolate other quantities as well as displacement due to the scarcity of c1 finite elements and their perceived computational cost in this context the lack of three dimensional c1 elements is of particular concern in this paper we present a new c1 hexahedral element which to the best of our knowledge is the first three dimensional c1 element ever constructed it is shown to pass the single element and patch tests and to give excellent rates of convergence in benchmark boundary value problems of gradient elasticity it is further shown that c1 elements are not necessarily more computationally expensive than alternative approaches and it is argued that they may be more efficient in providing good quality solutions copyright 2008 john wiley sons ltd

an advanced boundary element method for solving 2d and 3d static problems in mindlin s strain gradient theory of elasticity an advanced boundary element method bem for solving two 2d and three dimensional 3d problems in materials with microstructural effects is presented the analysis is performed in the context of mindlin s form ii gradient elastic theory the fundamental solution of the equilibrium partial differential equation is explicitly derived the integral representation of the problem consisting of two boundary integral equations one for displacements and the other for its normal derivative is developed the global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2d and 3d problems respectively representative 2d and 3d numerical examples are presented to illustrate the method demonstrate its accuracy and efficiency and assess the gradient effect on the response the importance of satisfying the correct boundary conditions in gradient elastic problems is illustrated with the solution of simple 2d problems copyright 2010 john wiley sons ltd

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

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

on the c1 continuous discretization of non linear gradient elasticity a comparison of nem and fem based on bernstein bezier patches in gradient elasticity the appearance of strain gradients in the free energy density leads to the need of c 1 continuous discretization m ethods in the present work the performances of c 1 finite elements and the c 1 natural element method nem are compared the triangular argyris and hsieh clough tocher finite elements are reparametrized in terms of the bernstein polynomials the quadrilateral bogner fox schmidt element is used in an isoparametric framework for which a preprocessing algorithm is presented additionally the c 1 nem is applied to non linear gradient elasticity several numerical examples are analyzed to compare the convergence behavior of the different methods it will be illustrated that the isoparametric elements and the nem show a significantly better performance than the triangular elements

boundary element analysis of mode i and mixed mode i and ii crack problems of 2 d gradient elasticity abstract a boundary element method is developed for fracture analysis of gradient elastic planar solids under static loading a simple version of mindlin s general theory of gradient elastic materials is employed and the two required boundary integral equations one for the displacement vector and the other for its normal derivative are presented use is made of the fundamental solution of the problem and this leads to a formulation that requires only a boundary discretization two representative numerical examples are presented to illustrate the method demonstrate its accuracy and efficiency and assess the gradient effect on the response the first deals with a mode i crack while the second with a mixed mode i ii crack for the second case the proposed method is used in conjunction with the method of subregions the method is employed with regular and special near the crack tip boundary elements the gradient effect consists of modifying both the displacement and the stress field around the crack tip and resulting in a response which is more physically acceptable than the one coming from the classical theory of elasticity

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