pointwise decay for the maxwell field on black hole space times abstract in this article we study the pointwise decay properties of solutions to the maxwell system on a class of nonstationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish peeling estimates as well as a t 4 rate of decay on compact regions for all the components of the maxwell tensor

blowup stability at optimal regularity for the critical wave equation abstract we establish strichartz estimates for the radial energy critical wave equation in 5 dimensions in similarity coordinates using these we prove the nonlinear asymptotic stability of the ode blowup in the energy space

strichartz estimates in similarity coordinates and stable blowup for the critical wave equation we establish strichartz estimates in similarity coordinates for the radial wave equation in three spatial dimensions with a time dependent self similar potential as an application we consider the critical wave equation and prove the asymptotic stability of the ode blowup profile in the energy space

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

estimation based approach for real time optimisation of uncertain nonlinear systems abstractin this paper the minimisation of an unknown but measurable cost function with uncertain dynamics is considered the drift term of the uncertain dynamical system and the gradient of the objective function are treated as unknown time varying parameters a novel estimation scheme based on almost invariant manifolds is proposed to estimate the time varying parameters a direct gradient based adaptive extremum seeking controller is designed to solve the uncertain optimisation problem this approach is shown to improve the transient performance of real time optimisation control systems

price s law on nonstationary space times abstract in this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t 3 local uniform decay rate price s law price 1972 54 for linear waves as a corollary we also prove price s law for certain small perturbations of the kerr metric this result was previously established by the second author in tataru 65 on stationary backgrounds the present work was motivated by the problem of nonlinear stability of the kerr schwarzschild solutions for the vacuum einstein equations which seems to require a more robust approach to proving linear decay estimates

strichartz and localized energy estimates for the wave equation in strictly concave domains we prove localized energy estimates for the wave equation in domains with a strictly concave boundary when homogeneous dirichlet or neumann conditions are imposed by restricting the solution to small frequency dependent space time collars of the boundary it is seen that a stronger gain in regularity can be obtained relative to the usual energy estimates mixed norm estimates of strichartz and square function type follow as a result using the energy estimates to control error terms which arise in a wave packet parametrix construction while the latter estimates are not new for dirichlet conditions the present approach provides an avenue for treating these estimates when neumann conditions are imposed the method also treats schr o dinger equations with time independent coefficients

global parametrices and dispersive estimates for variable coefficient wave equations in this article we consider variable coefficient time dependent wave equations in mathbb r times mathbb r n using phase space methods we construct outgoing parametrices and prove strichartz type estimates globally in time this is done in the context of c 2 metrics which satisfy a weak asymptotic flatness condition at infinity

dispersive estimates for principally normal pseudodifferential operators the aim of these notes is to describe some recent re sults concerning dispersive estimates for principally normal pseu dodifferential operators the main motivation for this comes from unique continuation problems such estimates can be used to prove l q carleman inequalities which in turn yield unique continuation results for various partial differential operators with rough poten tials dispersive estimates are l q estimates for nonelliptic partial differ ential operators which are a consequence of the decay properties of their fundamental solutions these decay properties follow from spa tial spreading of the singularities of the solutions since solutions prop agate in directions conormal to the characteristic set of the operator this spreading can be related to nonzero curvatures of the characteristic set dispersive estimates for constant coefficient operators are closely related to the restriction theorem in harmonic analysis various types of dispersive estimates are known to be true for op erators such as the wave operator the schrodinger operator and the linear kdv see ginibre velo 4 keel tao 11 they have proved to be useful in the study of nonlinear problems as well as of problems with unbounded potentials more recently similar estimates have been obtained for wave op erators with variable coefficients beginning with the smooth case in kapitanskii 10 mockenhaupt seeger and sogge 14 up to operators with c 2 coefficients in smith 15 and tataru 21 23 similar results were obtained for the schrodinger equation in staffilani tataru 19 c 2 coefficients and in burq gerard tzvetkov 1 smooth coeffic ients in the variable coefficient elliptic case one should also mention sogge s l q

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

on global strichartz estimates for non trapping metrics abstract we prove global strichartz estimates with spectral cutoff on the low frequencies for non trapping metric perturbations of the schrodinger equation posed on the euclidean space

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

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

parametrices and dispersive estimates for schrodinger operators with variable coefficients in this article we consider variable coefficient time dependent schr odinger evolutions in bbb r n using phase space methods we construct outgoing parametrices and prove strichartz type estimates globally in time this is done in the context of c 2 metrics which satisfy a weak asymptotic flatness condition at infinity

local decay of waves on asymptotically flat stationary space times author s tataru daniel abstract in this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t 3 local uniform decay rate for linear waves this work was motivated by open problems concerning decay rates for linear waves on schwarzschild and kerr backgrounds where such a decay rate has been conjectured by r price our results apply to both of these cases

strichartz estimates and local smoothing estimates for asymptotically flat schrodinger equations abstract in this article we study global in time strichartz estimates for the schrodinger evolution corresponding to long range perturbations of the euclidean laplacian this is a natural continuation of a recent article d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 of the third author where it is proved that local smoothing estimates imply strichartz estimates by d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 the local smoothing estimates are known to hold for small perturbations of the laplacian here we consider the case of large perturbations in three increasingly favorable scenarios i without non trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term ii with non trapping assumptions we prove global estimates modulo a lower order spatially localized error term and iii for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum

sharp low frequency resolvent estimates on asymptotically conical manifolds on a class of asymptotically conical manifolds we prove two types of low frequency estimates for the resolvent of the laplace beltrami operator the first result is a uniform l 2 rightarrow l 2 bound for scal r 1 delta g z 1 scal r 1 when mbox re z is small with the optimal weight scal r 1 the second one is about powers of the resolvent for any integer n we prove uniform l 2 rightarrow l 2 bounds for scal epsilon r n epsilon 2 delta g z n scal epsilon r n when mbox re z belongs to a compact subset of 0 infty and 0 epsilon ll 1 these results are obtained by proving similar estimates on a pure cone with a long range perturbation of the metric at infinity

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

smoothing properties and retarded estimates for some dispersive evolution equations smoothing properties in the form of space time integrability properties play an important role in the study of dispersive evolution equations a number of them follow from a combination of general arguments and specific estimates we present a general formulation which makes the separation between the two types of ingredients as clear as possible and we illustrate it with the examples of the schrodinger equation of the wave equation and of a class of 1 1 dimensional equations related to the benjamin ono equation of special interest for the cauchy problem are retarded estimates expressed in terms of those properties we derive a number of such estimates associated with the last example and we mention briefly an application of those estimates to the cauchy problem for the generalized benjamin ono equation

the glassey conjecture for nontrapping obstacles abstract we verify the 3 dimensional glassey conjecture for exterior domain m g where the metric g is asymptotically euclidean provided that certain local energy assumption is satisfied the radial glassey conjecture exterior to a ball is also verified for dimension three or higher the local energy assumption is satisfied for many important cases including exterior domain with nontrapping obstacles and flat metric exterior domain with star shaped obstacle and small asymptotically euclidean metric as well as the nontrapping asymptotically euclidean manifolds r n g

wavelet and receiver operating characteristic analysis of heart rate variability multiresolution wavelet analysis has been used to study the heart rate variability in two classes of patients with different pathological conditions the scale dependent measure of thurner et al was found to be statistically significant in discriminating patients suffering from hypercardiomyopathy from a control set of normal subjects we have performed receiver operating characteristc roc analysis and found the roc area to be a useful measure by which to label the significance of the discrimination as well as to describe the severity of heart dysfunction

the strauss conjecture on kerr black hole backgrounds we examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the strauss conjecture on the schwarzschild and kerr with small angular momentum black hole backgrounds the key estimates are a class of weighted strichartz estimates which are used near infinity where the metrics can be viewed as small perturbations of the minkowski metric and a localized energy estimate on the black hole background which handles the behavior in the remaining compact set

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

microlocal dispersive smoothing for the schrodinger equation this paper establishes a connection between the microlocal smoothness of solutions of the initial value problem for schrodinger s equation and the global behavior of bicharacteristics of the principal symbol in particular localized initial data gives rise to solutions which are microlocally smooth at all points which are not trapped backwards by the bicharacteristic flow the origin of the phenomenon is in the dispersive nature of the equation the results imply microlocal regularity properties of the fundamental solution 1995 john wiley sons inc