Space Of Initial Data Research Articles

Almost Sure Scattering for the One Dimensional Nonlinear Schrödinger Equation

We consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree p > 1 p>1 . On compact manifolds many probability measures are invariant by the flow of the linear Schrödinger equation (e.g. Wiener measures), and it is sometimes possible to modify them suitably and get invariant (Gibbs measures) or quasi-invariant measures for the non linear problem. On R d \mathbb {R}^d , the large time dispersion shows that the only invariant measure is the δ \delta measure on the trivial solution u = 0 u =0 , and the good notion to track is whether the non linear evolution of the initial measure is well described by the linear (nontrivial) evolution. This is precisely what we achieve in this work. We exhibit measures on the space of initial data for which we describe the nontrivial evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. Actually, we give precise (and optimal) bounds on the Radon–Nikodym derivatives of these measures with respect to each other and we characterise their L p L^p regularity. We deduce from this precise description the global well-posedness of the equation for p > 1 p>1 and scattering for p > 3 p>3 (actually even for 1 > p ≤ 3 1>p \leq 3 , we get a dispersive property of the solutions and exhibit an almost sure polynomial decay in time of their L p + 1 L^{p+1} norm). To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties or decay) for the nonlinear evolution.

Unbounded Sturm attractors for quasilinear parabolic equations

AbstractWe analyse the asymptotic dynamics of quasilinear parabolic equations when solutions may grow up (i.e. blow up in infinite time). For such models, there is a global attractor which is unbounded and the semiflow induces a nonlinear dynamics at infinity by means of a Poincaré projection. In case the dynamics at infinity is given by a semilinear equation, then it is gradient, consisting of the so-called equilibria at infinity and their corresponding heteroclinics. Moreover, the diffusion and reaction compete for the dimensionality of the induced dynamics at infinity. If the equilibria are hyperbolic, we explicitly prove the occurrence of heteroclinics between bounded equilibria and/or equilibria at infinity. These unbounded global attractors describe the space of admissible initial data at event horizons of certain black holes.

Exact controllability for a Rayleigh beam with piezoelectric actuator

In this paper, exact controllability problem for a Rayleigh beam with piezoelectric actuator is considered. Controllability results show that the space of controllable initial data depends on the regularity of the control and the location of the actuator. Two different spaces of control, L2(0,T) and (H1(0,T))′, correspond to two different controllability properties, L2-controllability and (H1)′-controllability, respectively. The approach to prove controllability results is based on the Hilbert Uniqueness Method. Some non-controllability results are also obtained. In particular, non-controllability in short control time is studied by using Riesz basis property of exponential family in L2(0,T). Finally, minimal control time for exact controllability is obtained.

Paracausal deformations of Lorentzian metrics and Møller isomorphisms in algebraic quantum field theory

Given a pair of normally hyperbolic operators over (possibnly different) globally hyperbolic spacetimes on a given smooth manifold, the existence of a geometric isomorphism, called Møller operator, between the space of solutions is studied. This is achieved by exploiting a new equivalence relation in the space of globally hyperbolic metrics, called paracausal relation. In particular, it is shown that the Møller operator associated to a pair of paracausally related metrics and normally hyperbolic operators also intertwines the respective causal propagators of the normally hyperbolic operators and it preserves the natural symplectic forms on the space of (smooth) initial data. Finally, the Møller map is lifted to a *-isomorphism between (generally off-shell) CCR-algebras. It is shown that the Wave Front set of a Hadamard bidistribution (and of a Hadamard state in particular) is preserved by the pull-back action of this *-isomorphism.

Peeling-off behavior of wave equation in the Vaidya spacetime

We study the peeling for the wave equation on the Vaidya spacetime following the approach developed by Mason and Nicolas in Mason–Nicolas 2009. The idea is to encode the regularity at null infinity of the rescaled field, characterized by Sobolev-type norms, in terms of corresponding function spaces of initial data. All function spaces are obtained from energy fluxes associated with an observer constructed from the Morawetz vector field on Minkowski spacetime. We combine conformal techniques and energy estimates to obtain the optimal classes of initial data ensuring a given regularity of the rescaled field. The classes of data are equivalent to those obtained on Minkowski and Schwarzschild spacetimes in that they impose the same decay at infinity and regularity.

Estimates for the Heat Flow in Optimal Spaces of Unbounded Initial Data in mathbb {R}^{{d}} and Applications to the Ornstein–Uhlenbeck Semigroup

In this paper, we improve some estimates from [7] for solutions of the heat equation with initial conditions that are large ‘at infinity’ and employ them to obtain suitable estimates on the solutions of the Ornstein–Uhlenbeck semigroup [solutions of u_t-Delta u=alpha x cdot nabla u] for the same class of unbounded data.

Boundary dynamics in competing critical black hole formation

Expanding upon our previous study of competing critical phenomena in black hole formation, we numerically investigate the behavior of dominant exponents across the boundary separating asymptotically dispersing and collapsing regions in a two-dimensional configuration space of initial data. We find that across the Type II boundary section the dominant exponent remains constant, equal to the reciprocal of Choptuik's well-known quasi-universal value, whereas across the Type I section the exponent noticeably varies. We postulate that this change reflects the existence of a third critical solution in addition to the two primary competing solutions, possibly another member of the family of metastable soliton stars constituting the Type I attractor.

Local smoothing and Hardy spaces for Fourier integral operators

We show that the Hardy spaces for Fourier integral operators form natural spaces of initial data when applying ℓp-decoupling inequalities to local smoothing for the wave equation. This yields new local smoothing estimates which, in a quantified manner, improve the bounds in the local smoothing conjecture on Rn for p≥2(n+1)/(n−1), and complement them for 2<p<2(n+1)/(n−1). These estimates are invariant under application of Fourier integral operators, and they are essentially sharp.

Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence* *The second author is supported by the VIDI subsidy 639.032.427 of the Netherlands Organisation for Scientific Research (NWO).

In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type. This first part is on local existence and well-posedness. A second part in preparation is on blow-up criteria and regularization. Our theory is formulated in an L p -setting, and because of this we can deal with nonlinearities in a very efficient way. Applications to several concrete problems and their quasilinear variants are given. This includes Burgers’ equation, the Allen–Cahn equation, the Cahn–Hilliard equation, reaction–diffusion equations, and the porous media equation. The interplay of the nonlinearities and the critical spaces of initial data leads to new results and insights for these SPDEs. The proofs are based on recent developments in maximal regularity theory for the linearized problem for deterministic and stochastic evolution equations. In particular, our theory can be seen as a stochastic version of the theory of critical spaces due to Prüss–Simonett–Wilke (2018). Sharp weighted time-regularity allow us to deal with rough initial values and obtain instantaneous regularization results. The abstract well-posedness results are obtained by a combination of several sophisticated splitting and truncation arguments.

PROPERTIES OF THE EQUATION OF HEAT CONDUCTION WITH DISSIPATION SOLUTIONS

This paper investigates the properties of the solutions of the equation of heat conduction with dissipation, which is associated with a harmonic oscillator - the operator $-d^2/dx^2 + x^2$, $x\in \mathbb{R}$ (non-negative and self-adjoint in $L_2(\mathbb{R})$). An explicit form of the function is given, which is analogous to the fundamental solution of the Cauchy problem for the heat conduction equation. A formula that describes all infinitely differentiable (with respect to the variable $x$) solutions of such an equation was found, well-posedness of the Cauchy problem for the heat conduction equation with dissipation with the initial function, which is an element of the space of generalized functions $(S_{1/2}^{1/2})'$, is established. It is established that $(S_{1/2}^{1/2})'$ is the "maximum" space of initial data of the Cauchy problem, for which the solutions are infinite functions differentiable by spatial variable. The main means of research are formal Hermite series, which are identified with linear continuous functionals defined on $S_{1/2}^{1/2}$.

Quasinormal Modes in Extremal Reissner\u2013Nordstr\xf6m Spacetimes

We present a new framework for characterizing quasinormal modes (QNMs) or resonant states for the wave equation on asymptotically flat spacetimes, applied to the setting of extremal Reissner–Nordström black holes. We show that QNMs can be interpreted as honest eigenfunctions of generators of time translations acting on Hilbert spaces of initial data, corresponding to a suitable time slicing. The main difficulty that is present in the asymptotically flat setting, but is absent in the previously studied asymptotically de Sitter or anti de Sitter sub-extremal black hole spacetimes, is that L^2-based Sobolev spaces are not suitable Hilbert space choices. Instead, we consider Hilbert spaces of functions that are additionally Gevrey regular at infinity and at the event horizon. We introduce L^2-based Gevrey estimates for the wave equation that are intimately connected to the existence of conserved quantities along null infinity and the event horizon. We relate this new framework to the traditional interpretation of quasinormal frequencies as poles of the meromorphic continuation of a resolvent operator and obtain new quantitative results in this setting.

Hamiltonian charges in the asymptotically de Sitter spacetimes

We generalize a notion of ‘conserved’ charges given by Wald and Zoupas to the asymptotically de Sitter spacetimes. Surprisingly, our construction is less ambiguous than the one encountered in the asymptotically flat context. An expansion around exact solutions possessing Killing vectors provides their physical meaning. In particular, we discuss a question of how to define energy and angular momenta of gravitational waves propagating on Kottler and Carter backgrounds. We show that obtained expressions have a correct limit as Λ → 0. We also comment on the relation between this approach and the one based on the canonical phase space of initial data at ℐ+.

Weak-strong uniqueness in weighted L2 spaces and weak suitable solutions in local Morrey spaces for the MHD equations

We consider here the magneto-hydrodynamics (MHD) equations on the whole space. For the 3D case, in the setting of the weighted L2 spaces we obtain a weak-strong uniqueness criterion provided that the velocity field and the magnetic field belong to a fairly general multipliers space. On the other hand, we study the local and global existence of weak suitable solutions for intermittent initial data, which is characterized through a local Morrey space. This large initial data space was also exhibit in a contemporary work [4] in the context of 3D Navier-Stokes equations. Finally, we make a discussion on the local and global existence problem in the 2D case.

The heat flow in an optimal Fréchet space of unbounded initial data in [formula omitted

In this paper we show that solutions of the heat equation that are given in terms of the heat kernel define semigroups on the family of Fréchet spaces L0p(Rd), the intersection (over all ε>0) of the spaces Lεp(Rd) of functions such that ∫Rde−ε|x|2|f(x)|pdx<∞. These spaces consist of functions that are ‘large at infinity’, and L01(Rd) is the maximal space in which one can use the heat kernel to obtain globally-defined solutions of the heat equation. We prove suitable estimates from L0p(Rd) into L0q(Rd), q≥p, for these semigroups.We then consider the heat semigroup posed in spaces that are dual to these spaces of functions, namely the spaces L−εp(Rd) of very-rapidly decreasing functions such that ∫Rdeε|x|2|f(x)|pdx<∞. We show that (Lpεp(Rd))′=L−qεq(Rd) (with 1<p<∞ and (p,q) conjugate), and that the heat flow on Lεp(Rd) is the adjoint of the flow on L−δq(Rd) for an appropriate (time-dependent) choice of δ.

Forecasting S&amp;P 500 spikes: an SVM approach

In this study, we focus on forecasting long-tail events of the SP in our case, we use the returns that fall outside a two-standard deviations band. However, instead of simply using the unconditional overall standard deviation, in this paper we employ a GARCH (p,q) model to derive the conditional standard deviation of the returns. This is a more appropriate measure of immediate risk to market participants than the overall series’ unconditional standard deviation. Traditional forecasting models that rely on statistical analysis and traditional econometrics, assume that the returns follow some typical underlying distribution. These models usually fail to successfully and efficiently accommodate price spikes especially when it comes to forecasting. In our study, we use the atheoretical and data-driven Support Vector Machines methodology from the area of Machine Learning. This forecasting approach does not require any initial assumptions on the distribution of the data but rather exploits patterns that may be inhibited in the initial data space. These patterns may become more apparent and exploitable in the resulting feature space. We use 1860 daily observations from 01/01/2009 to 22/01/2017. Our overall optimum forecasting model achieved a 70.69% forecasting accuracy for the spikes and 73.25% for non-spikes.

Investment Boolean problem with Savage risk criteria under uncertainty

AbstractThe portfolio theory is used to formulate a multicriteria investment Boolean escaped gain minimization problem for searching all extreme portfolios. Stability aspects of this set against perturbed parameters of minimax Savage criteria are studied. We give lower and upper estimates for the stability radius for arbitrary Hölder norms on the three-dimensional space of initial data.

Exterior Energy Bounds for the Critical Wave Equation Close to the Ground State

By definition, the exterior asymptotic energy of a solution to a wave equation on $\mathbb{R}^{1+N}$ is the sum of the limits as $t\to \pm\infty$ of the energy in the the exterior $\{|x|>|t|\}$ of the wave cone. In our previous work (JEMS 2012, arXiv:1003.0625), we have proved that the exterior asymptotic energy of a solution of the linear wave equation in odd space dimension $N$ is bounded from below by the conserved energy of the solution. In this article, we study the analogous problem for the linear wave equation with a potential \begin{equation} \label{abstractLW} \tag{*} \partial_t^2u+L_Wu=0,\quad L_W:=-\Delta -\frac{N+2}{N-2}W^{\frac{4}{N-2}} \end{equation} obtained by linearizing the energy critical wave equation at the ground-state solution $W$, still in odd space dimension. This equation admits nonzero solutions of the form $A+tB$, where $L_WA=L_WB=0$ with vanishing asymptotic exterior energy. We prove that the exterior energy of a solution of \eqref{abstractLW} is bounded from below by the energy of the projection of the initial data on the orthogonal complement of the space of initial data corresponding to these solutions. This will be used in a subsequent paper to prove soliton resolution for the energy-critical wave equation with radial data in all odd space dimensions. We also prove analogous results for the linearization of the energy-critical wave equation around a Lorentz transform of $W$, and give applications to the dynamics of the nonlinear equation close to the ground state in space dimensions $3$ and $5$.

Application of the mol2vec Technology to Large-size Data Visualization and Analysis.

Generative Topographic Mapping (GTM) is a dimensionality reduction method, which is widely used for both data visualization and structure-activity modeling. Large dimensionality of the initial data space may require significant computational resources and slow down the GTM construction. Therefore, it may be meaningful to reduce the number of descriptors used for encoding molecular structures. The Principal Component Analysis (PCA), a standard preprocessing tool, suffers from the information loss upon the dimensionality reduction. As an alternative, we propose to use substructure vector embedding provided by the mol2vec technique. In addition to the data dimensionality reduction, this technology also accounts for proximity of substructures in molecular graphs. In this study, dimensionality of large descriptor spaces of ISIDA fragment descriptors or Morgan fingerprints were reduced using either the PCA or the mol2vec method. The latter significantly speeds up GTM training without compromising its predictive power in bioactivity classification tasks.

Existence of Suitable Weak Solutions to the Navier–Stokes Equations for Intermittent Data

We obtain a global existence result for the three-dimensional Navier-Stokes equations with a large class of data allowing growth at spatial infinity. Namely, we show the global existence of suitable weak solutions when the initial data belongs to the weighted space $\mathring M^{2,2}_{\mathcal C}$ introduced in [Z. Bradshaw and I. Kukavica, Existence of suitable weak solutions to the Navier-Stokes equations for intermittent data, J. Math. Fluid Mech. to appear]. This class is strictly larger than currently available spaces of initial data for global existence and includes all locally square integrable discretely self-similar data. We also identify a sub-class of data for which solutions exhibit eventual regularity on a parabolic set in space-time.

Bianchi models with a free massless scalar field: invariant sets and higher symmetries

We scrutinize the overall structure of the space of cosmological models of Bianchi type I–VIIh that contain a free massless scalar field, with a spatially homogeneous gradient that generally breaks isotropy, in addition to a standard perfect fluid. Specifically, state space is written as a union of disjoint invariant sets, each corresponding to a particular cosmological model that is classified with respect to the Bianchi type and the matter content. Subsets corresponding to higher-symmetry models, including all locally rotationally symmetric models and FLRW models, and models with a shear-free normal congruence, are also derived and classified. For each model the dimension (d) of the space of initial data is given, after fixing the orientation of the orthonormal frame uniquely relative to matter anisotropies and geometrical anisotropies.