global regularity for the maxwell klein gordon equation with small critical sobolev norm in high dimensions we show that in dimensions n 6 one has global regularity for the maxwell klein gordon equations in the coulomb gauge provided that the critical sobolev norm open image in new window of the initial data is sufficiently small these results are analogous to those recently obtained for the high dimensional wave map equation 17 7 14 12 but unlike the wave map equation the coulomb gauge non linearity cannot be iterated away directly we shall use a different approach proving strichartz estimates for the covariant wave equation this in turn will be achieved by use of littlewood paley multipliers and a global parametrix for the covariant wave equation constructed using a truncated microlocalized cronstrom gauge

a qualitative analysis of two and three step methods for stable second order systems a variety of two and three step direct integration methods have been proposed for solving stable systems of linear second order ordinary differential equations we will assess their suitability using as our main criterion their ability to reproduce stable numerical solutions for stable positive definite and symmetric differential systems in addition we determine which of these methods have the ability to reproduce the weak and strong damping properties associated with these differential systems next we investigate their suitability for solving unsymmetric linear gyroscopic systems and finally we show that for a wider class of unsymmetric but stable differential systems only a very restricted class of methods may be stable through this investigation we isolate all the potentially useful two step methods as a subset of the newmark methods similarly we identify a class of useful second order three step methods and this latter investigation provides new insight into the popular houbolt and wilson

higher order single step fully discrete approximations for second order hyperbolic equations with time dependent coefficients error estimates are proved for finite element approximations to the solution of an initial boundary value problem for a second order hyperbolic partial differential equation with time dependent coefficients optimal order rates of convergence in l 2 are shown for single step fully discrete schemes the order nu of the single step schemes is greater than or equal to one and less than or equal to four a class of iterative methods for approximately solving the fully discrete equations is analyzed and optimal order rates of convergence are also obtained these methods avoid the repeated work of new matrix factorizations at each time level

reduced thin sandwich equations on manifolds euclidean at infinity and on closed manifolds existence and multiplicity the reduced thin sandwich equations rtse appear within wheeler s thin sandwich approach toward the einstein constraint equations ece of general relativity it is known that these equations cannot be well posed in general but on closed manifolds sufficient conditions for well posedness have been established in particular it has been shown that the rtse are well posed in a neighborhood of umbilical solutions of the constraint equations without conformal killing fields in this paper we will analyze such a set of equations on the manifolds euclidean at infinity in a neighborhood of asymptotically euclidean ae solutions of the ece the main conclusion in this direction is that on ae manifolds admitting a yamabe positive metric the solutions of the rtse parameterize an open subset in the space of solutions of the ece in addition we show that in the case of closed manifolds these equations are well posed around umbilical solutions of the ece admitting killing fields and present some relevant examples finally it will be shown that in the set of umbilical solutions of the vacuum ece on closed manifolds the rtse are generically well posed

higher derivative explicit one step methods for non linear dynamic problems part i design and theory a new higher order explicit one step scheme is presented for use in the solution of a certain class of non linear problems in structural dynamics the generalized scheme covers some recently developed multistep algorithms as well as some new one step methods with improved numerical properties a weighted residual approach is used which contains new aspects for design of single step algorithms accuracy and stability are investigated for linear and non linear systems algorithms with higher accurary than the central difference method are developed and optimized to minimize the reduction in the stability limit

improving stability domains of the implicit higher order accuracy method the implicit higher order accuracy ihoa time integration family has a high order of accuracy however its stability domains are very restrictive by improving the stability conditions the larger time steps can be utilized in this study two parameters are introduced in the displacement and velocity extrapolations of ihoa in order to find the optimum values of the proposed parameters the dissipation and the dispersion are investigated three conditionally stable versions of the second third and forth order algorithms are found it is shown that these formulations have larger stability domains than the previous ones furthermore the suggested strategies give responses that are more accurate copyright 2011 john wiley sons ltd

recent progress in extrapolation methods for ordinary differential equations the paper is intended to give a survey on the state of the art of extrapolation methods for initial value problems in ordinary differential equation systems all basic discretizations which are su

a unified set of single step algorithms part 3 the beta m method a generalization of the newmark scheme introduced herein is a generalization of newmark s time marching integration scheme called the m method like the sspj method introduced in parts 1 and 2 of this series the m method provides a gcneral single step scheme applicable to any set of initial value problems the method is specialized by specifying the method order m along with rn integration parameters 0 1 m 1 for a particular choice of m the integration parameters provide a subfamily of methods which control accuracy and stability as well as offering options for explicit or implicit algorithms for the most part attention is focused on the application to structural dynamic equations most well known methods e g newmark wilson houbolt etc are shown to be special cases within the m family hence one computationally efficient and surprisingly simple algorithm unifies old and new methods stability and accuracy analyses are presented for method orders m 2 3 and 4 to determine optimal parameters for implicit and explicit schemes along with numerical verification

second derivative methods applied to implicit first and second order systems a wide variety of methods have been proposed for solving implicit second order systems of differential equations such as arise in structural and mechanical vibration analyses here we discuss a class of methods based on second derivative formulae some of which have previously been used by enright 1 we show them to have desirable analytic and implementation properties we relate these methods to others recently proposed by brusa and nigro and by serbin showing that we obtain algorithms which have all the desirable properties of their methods and which can also be extended to classes of practical problems which were not considered by these other authors since implicit first order systems are also important in practice for example in finite element analyses of heat conduction problems we demonstrate the efficient implementation of the second derivative formulae in this case too

variable order variable step algorithms for second order systems part 2 the codes we develop a new class of multistage algorithms for second order systems of ordinary differential equations ode s based on methods by zienkiewicz et al 13 our new methods are designed to provide a set of efficient methods which can be used with local error estimators based on embedding techniques in a companion paper we provide a set of subroutines implementing the algorithms derived and a discussion of numerical experience with the resulting codes

strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation the aim of this article is threefold first we use the fbi transform to set up a calculus for partial differential operators with nonsmooth coefficients next this calculus allows us to prove strichartz type estimates for the wave equation with nonsmooth coefficients finally we use these strichartz estimates to improve the local theory for second order nonlinear hyperbolic equations 1 introduction the fbi transform is in a way similar to the complex fourier transform in that for each function in rn it provides a representation as a holomorphic function in r2n however in the case of the fbi transform we can identify naturally r2n with the phase space t rn for a pseudodifferential operator with smooth symbol acting on functions in rn one can produce by conjugation a corresponding formal series acting on func tions in r2n9 for which the first term is exactly the multiplication by the symbol this series converges and has a nice representation in the weyl calculus provided that the symbol of the operator is analytic this is how the fbi transform has been used in the study of partial differential operators with analytic coefficients see 12 13 where this machinery is developed here in a way we do the opposite we look at operators with nonsmooth coefficients approximate the conjugated operator by a partial sum of the formal series and prove error estimates in the simplest case the approximate conjugate operator is exactly the multiplication by the symbol this is also related to the cordoba fefferman wave packet transform in 3 in the third section we use the error estimates to reduce the strichartz estimates for second order hyperbolic operators with nonsmooth coefficients to weighted lp lq l2 estimates for the fbi transform these in turn can be proved in the usual fashion using appropriate oscillatory integral esti mates in the last part of the article we explain how one can use the new strichartz type estimates to improve the local theory for nonlinear hyperbolic equations beyond the classical setup these results are not sharp and will be improved in subsequent articles

counterexamples to local existence for semi linear wave equations in this paper we study how much regularity of initial data is needed to ensure existence of a local solution to a semi linear wave equation we give counterexamples to local existence for the typical model equations the counterexamples we construct are sharp i e one does have a local solution if the data has slightly more regularity

a unified set of single step algorithms part 1 general formulation and applications a general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented this algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities in many cases it presents a computationally advantageous form over conventional procedures this is particularly so when compared with the newmark algorighm and its variants

a further look at newmark houbolt etc time stepping formulae this paper looks at some well known methods for the numerical integration of the vibration equation from the point of view of various groupings of the independent equations involved new forms are given of the general three parameter method introduced by zienkiewicz 1 some new one step methods are introduced these are of great practical importance because of the ease with which the size of time step can be changed

strichartz estimates for second order hyperbolic operators with nonsmooth coefficients ii in an earlier work we have proved some estimates of strichartz type for second order hyperbolic operators with c s coefficients for 0 s 1 here we improve these estimates and extend the range of s to 0 s 2 in particular for s 2 we obtain the full strichartz estimates

causal geometry of einstein vacuum spacetimes with finite curvature flux one of the central difficulties of settling the l2 bounded curvature conjecture for the einstein vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity in this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of enstein vacuum spacetimes can be controlled in terms of initial data and the total curvature flux through the hypersurface

properties of multistep formulas intended for a class of second order odes we analyze the stability and order properties of multistep formulas suitable for second order odes of the form my cy ky r t these formulas are actually formula pairs where one formula is used to obtain an approximation to y and the other to obtain an approximation to y we show how to transform a k step formula pair into a 2k step single formula whose stability and order properties can be more easily studied we present conditions that ensure that the formula pair is stable at zero and infinity and we demonstrate that the stability properties of a formula pair can differ substantially from those of its component formulas the order analysis provides necessary and sufficient conditions on the coefficients of a formula pair to obtain a local truncation error of order p 2 these conditions are dependent upon the order properties of the component formulas consequently given the order of the component formulas the order of the formula pair is easily determined

equations d ondes quasilineaires et estimations de strichartz dans cet article notre but est de resoudre des equations d ondes quasilineaires pour des donnees initiales moins regulieres que ce qu imposent les methodes d energie ceci impose de demontrer des estimees de type strichartz pour des operateurs d ondes a coefficients seulement lipschitziens

solutions of the einstein constraint equations with apparent horizon boundaries we construct asymptotically euclidean solutions of the vacuum einstein constraint equations with an apparent horizon boundary condition specifically we give sufficient conditions for the constant mean curvature conformal method to generate such solutions the method of proof is based on the barrier method used by isenberg for compact manifolds without boundary suitably extended to accommodate semilinear boundary conditions and low regularity metrics as a consequence of our results for manifolds with boundary we also obtain improvements to the theory of the constraint equations on asymptotically euclidean manifolds without boundary

smooth deformations of singular contractions of class vii surfaces we consider normal compact surfaces y obtained from a minimal class vii surface x by contraction of a cycle c of r rational curves with c 2 0 our main result states that if the obtained cusp is smoothable then y is globally smoothable the proof is based on a vanishing theorem for h 2 theta y if r b 2 x any smooth small deformation of y is rational and if r b 2 x i e when x is a half inoue surface any smooth small deformation of y is an enriques surface the condition the cusp is smoothable in our main theorem can be checked in terms of the intersection numbers of the cycle using the looijenga conjecture which has recently become a theorem therefore this is a decidable condition we prove that this condition is always satisfied if r b 2 x leqslant 11 therefore the singular surface y obtained by contracting a cycle c of r rational curves in a minimal class vii surface x with r b 2 x leqslant 11 is always smoothable by rational surfaces the statement holds even for unknown class vii surfaces

sharp local well posedness results for the nonlinear wave equation this article is concerned with local well posedness of the cauchy problem for second order quasilinear hyperbolic equations with rough initial data the new results obtained here are sharp in low dimension

the bounded l 2 curvature conjecture this is the main paper in a sequence in which we give a complete proof of the bounded l 2 curvature conjecture more precisely we show that the time of existence of a classical solution to the einstein vacuum equations depends only on the l 2 norm of the curvature and a lower bound on the volume radius of the corresponding initial data set we note that though the result is not optimal with respect to the scaling of the einstein equations it is nevertheless critical with respect to its causal geometry indeed l 2 bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries we note also that while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than 1 1 based on strichartz estimates were obtained in bahouri and chemin am j math 121 1337 1777 1999 imrn 21 1141 1178 1999 tataru am j math 122 349 376 2000 jams 15 2 419 442 2002 klainerman and rodnianski duke math j 117 1 1 124 2003 and optimized in klainerman and rodnianski ann math 161 1143 1193 2005 smith and tataru ann math 162 291 366 2005 the result we present here is the first in which the full structure of the quasilinear hyperbolic system not just its principal part plays a crucial role to achieve our goals we recast the einstein vacuum equations as a quasilinear so 3 1 valued yang mills theory and introduce a coulomb type gauge condition in which the equations exhibit a specific new type of null structure compatible with the quasilinear covariant nature of the equations to prove the conjecture we formulate and establish bilinear and trilinear estimates on rough backgrounds which allow us to make use of that crucial structure these require a careful construction and control of parametrices including l 2 error bounds which is carried out in szeftel parametrix for wave equations on a rough background i regularity of the phase at initial time arxiv 1204 1768 2012 parametrix for wave equations on a rough background ii construction of the parametrix and control at initial time arxiv 1204 1769 2012 parametrix for wave equations on a rough background iii space time regularity of the phase arxiv 1204 1770 2012 parametrix for wave equations on a rough background iv control of the error term arxiv 1204 1771 2012 as well as a proof of sharp strichartz estimates for the wave equation on a rough background which is carried out in szeftel sharp strichartz estimates for the wave equation on a rough background arxiv 1301 0112 2013 it is at this level that our problem is critical indeed any known notion of a parametrix relies in an essential way on the eikonal equation and our space time possesses barely the minimal regularity needed to make sense of its solutions

global well posedness for the maxwell klein gordon equation in 4 1 dimensions small energy we prove that the critical maxwell klein gordon equation on r4 1 is globally well posed for smooth initial data which are small in the energy norm this reduces the problem of global regularity for large smooth initial data to precluding concentration of energy