Local Energy Decay Estimate Research Articles

The effect of metric behavior at spatial infinity on pointwise wave decay in the asymptotically flat stationary setting

abstract: The current work considers solutions to the wave equation on asymptotically flat, stationary, Lorentzian spacetimes in $(1+3)$ dimensions. We investigate the relationship between the rate at which the geometry tends to flat and the pointwise decay rate of solutions. The case where the spacetime tends toward flat at a rate of $|x|^{-1}$ was studied by Tataru (2013), where a $t^{-3}$ pointwise decay rate was established. Here we extend the result to geometries tending toward flat at a rate of $|x|^{-\kappa}$ and establish a pointwise decay rate of $t^{-\kappa-2}$ for $\kappa\in\Bbb{N}$ with $\kappa\ge 2$. We assume a weak local energy decay estimate holds, which restricts the geodesic trapping allowed on the underlying geometry. We use the resolvent to connect the time Fourier Transform of a solution to the Cauchy data. Ultimately the rate of pointwise wave decay depends on the low frequency behavior of the resolvent, which is sensitive to the rate at which the background geometry tends to flat.

L2-blowup estimates of the wave equation and its application to local energy decay

We consider the Cauchy problems in [Formula: see text] for the wave equation with a weighted [Formula: see text]-initial data. We derive sharp infinite time blowup estimates of the [Formula: see text]-norm of solutions in the case of [Formula: see text] and [Formula: see text]. Then, we apply it to the local energy decay estimates for [Formula: see text], which is not studied so completely when the [Formula: see text]th moment of the initial velocity does not vanish. The idea to derive them is strongly inspired from a technique used in [R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differ. Equ. 257 (2014) 2159–2177; R. Ikehata and M. Onodera, Remarks on large time behavior of the [Formula: see text]-norm of solutions to strongly damped wave equations, Differ. Integral Equ. 30 (2017) 505–520].

Sharp Decay for Teukolsky Equation in Kerr Spacetimes

In this work, we derive the global sharp decay, as both a lower and an upper bounds, for the spin pm {mathfrak {s}} components, which are solutions to the Teukolsky equation, in the black hole exterior and on the event horizon of a slowly rotating Kerr spacetime. These estimates are generalized to any subextreme Kerr background under an integrated local energy decay estimate. Our results apply to the scalar field ({mathfrak {s}}=0), the Maxwell field ({mathfrak {s}}=1) and the linearized gravity ({mathfrak {s}}=2) and confirm the Price’s law decay that is conjectured to be sharp. Our analyses rely on a novel global conservation law for the Teukolsky equation, and this new approach can be applied to derive the precise asymptotics for solutions to semilinear wave equations.

Pointwise decay for the wave equation on nonstationary spacetimes

This first article in a two-part series (the second article being [arXiv:2205.13197]) assumes a weak form of the integrated local energy decay estimate and proves that solutions to the linear wave equation with variable coefficients in R1+3, first-order terms, and a potential decay at a rate depending on how rapidly the vector fields of these functions decay at spatial infinity. Roughly speaking, given metric perturbation terms and first-order terms whose vector fields decay at least as rapidly as the rate r−1−a, and potentials whose vector fields decay at least as rapidly as the rate r−2−a, our results imply that |ZJϕ(t,x)|≤Ct−2−a for bounded |x|, where Z denotes vector fields and J is a multi-index. We expect that this is a sharp decay rate. We prove results for both stationary and nonstationary metrics and coefficients. The proof uses integrated local energy decay to achieve an initial decay rate, and then uses an iteration involving the one-dimensional reduction (due to the positivity of the fundamental solution of the wave equation in three spatial dimensions) to achieve the full decay rate. For second-order perturbations of the Minkowski metric with coefficients that are not necessarily spherically symmetric, we analyze these coefficients' decay rates in a novel way to obtain a higher rate of pointwise decay for the solution than might be expected, assuming only weak pointwise decay on up to two time derivatives of the coefficients (and vector fields thereof); the latter weak decay is satisfied if, for example, the metric is stationary.

Generic Transporters for the Linear Time-Dependent Quantum Harmonic Oscillator on ℝ

Abstract In this paper we consider the linear, time-dependent quantum Harmonic Schrdinger equation ${\textrm {i}} \partial _t u= \frac {1}{2} ( - \partial _x^2 + x^2) u + V(t, x, D)u$, $x \in \mathbb {R}$, where $V(t,x,D)$ is classical pseudodifferential operator of order 0, self-adjoint, and $2\pi $ periodic in time. We give sufficient conditions on the principal symbol of $V(t,x,D)$ ensuring the existence of solutions displaying infinite time growth of Sobolev norms. These conditions are generic in the Fréchet space of symbols. This shows that generic, classical pseudodifferential, $2\pi $-periodic perturbations provoke unstable dynamics. The proof builds on the results of [36] and it is based on pseudodifferential normal form and local energy decay estimates. These last are proved exploiting Mourre’s positive commutator theory.

A Morawetz inequality for water waves

Author(s): Alazard, Thomas; Ifrim, Mihaela; Tataru, Daniel | Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit.

Asymptotic stability of small ground states for NLS under random perturbations

We consider the cubic Schrödinger (NLS) equation on the Euclidean space perturbed by a short-range potential V . The presence of a negative simple eigenvalue for -\Delta+V gives rise to a curve of small and localized nonlinear ground states that yield some time-periodic solutions known to be asymptotically stable in the energy space. We study the persistence of these coherent states under rough perturbations. We shall construct a large measure set of small scaling-supercritical solutions below the energy space that display some asymptotic stability behaviors. The main difficulty is the need to handle the interactions of localized and dispersive terms in the modulation equations. To do so, we use a critical-weighted strategy to combine probabilistic nonlinear estimates in critical spaces based on U^p , V^q (controlling higher-order terms) with some local energy decay estimates (controlling lower-order terms). We also revisit in the perturbed setting the analysis of Bényi, Oh, and Pocovnicu (2015) on the probabilistic global well-posedness and scattering for small supercritical initial data. We use a distorted Fourier transform and semiclassical functional calculus to generalize probabilistic and bilinear Strichartz estimates.

Global behaviors of defocusing semilinear wave equations

In this paper, we investigate the global behaviors of solutions to defocusing semilinear wave equations in $\mathbb{R}^{1+d}$ with $d\geq 3$. We prove that in the energy space the solution verifies the integrated local energy decay estimates for the full range of energy subcritical and critical power. For the case when $p>1+\frac{2}{d-1}$, we derive a uniform weighted energy bound for the solution as well as inverse polynomial decay of the energy flux through hypersurfaces away from the light cone. As a consequence, the solution scatters in the energy space and in the critical Sobolev space for $p$ with an improved lower bound. This in particular extends the existing scattering results to higher dimensions without spherical symmetry.

Sharp decay estimates for massless Dirac fields on a Schwarzschild background

We consider the explicit asymptotic profile of massless Dirac fields on a Schwarzschild background. First, we prove for the spin s=±12 components of the Dirac field a uniform bound of a positive definite energy and an integrated local energy decay estimate from a symmetric hyperbolic wave system. Based on these estimates, we further show that these components have globally pointwise decay fv−3/2−sτ−5/2+s as both an upper and a lower bound outside the black hole, with function f finite and explicitly expressed in terms of the initial data and the coordinates. This establishes the validity of the conjectured Price's law for massless Dirac fields outside a Schwarzschild black hole.

Scattering for critical wave equations with variable coefficients

AbstractWe prove that solutions to the quintic semilinear wave equation with variable coefficients in ${{\mathbb {R}}}^{1+3}$ scatter to a solution to the corresponding linear wave equation. The coefficients are small and decay as $|x|\to \infty$, but are allowed to be time dependent. The proof uses local energy decay estimates to establish the decay of the $L^{6}$ norm of the solution as $t\to \infty$.

A Morawetz Inequality for Gravity-Capillary Water Waves at Low Bond Number

This paper is devoted to the 2D gravity-capillary water waves equations in their Hamiltonian formulation, addressing the general question of proving Morawetz inequalities. We continue the analysis initiated in our previous work, where we have established local energy decay estimates for gravity waves. Here we add surface tension and prove a stronger estimate with a local regularity gain, akin to the smoothing effect for dispersive equations. Our main result holds globally in time and holds for genuinely nonlinear waves, since we are only assuming some very mild uniform Sobolev bounds for the solutions. Furthermore, it is uniform both in the infinite depth limit and the zero surface tension limit.

Morawetz Estimate for Linearized Gravity in Schwarzschild

The equations governing the perturbations of the Schwarzschild metric satisfy the Regge–Wheeler–Zerilli–Moncrief system. Applying the technique introduced in Andersson and Blue (Ann Math 182(2):787–853, 2015), we prove an integrated local energy decay estimate for both the Regge–Wheeler and Zerilli equations. In these proofs, we use some constants that are computed numerically. Furthermore, we make use of the \(r^p\) hierarchy estimates (Dafermos and Rodnianski, in: Exner (ed) XVIth international congress on mathematical physics, World Scientic, London, pp 421–433, 2009; Schlue in Anal PDE 6:515–600, 2013) to prove that both the Regge–Wheeler and Zerilli variables decay as \(t^{-\frac{3}{2}}\) in fixed regions of r.

Uniform Energy Bound and Morawetz Estimate for Extreme Components of Spin Fields in the Exterior of a Slowly Rotating Kerr Black Hole I: Maxwell Field

This first part of the series treats the Maxwell equations in the exterior of a slowly rotating Kerr black hole. By performing a first-order differential operator on each extreme Newman-Penrose (N-P) scalar in a Kinnersley tetrad, the resulting equation and the Teukolsky master equation for the extreme N-P component are both in the form of an inhomogeneous \textquotedblleft{spin-weighted Fackerell-Ipser equation\textquotedblright} (SWFIE) and constitute a weakly coupled system. We first prove energy estimate and integrated local energy decay (Morawetz) estimate for this type of inhomogeneous SWFIE following the method in (Dafermos and Rodnianski in Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases $|a|\ll M$ or axisymmetry, 2010, arXiv:1010.5132), and then utilize these estimates to achieve both a uniform bound of a positive definite energy and a Morawetz estimate for the coupled system of each extreme N-P component. The same type of estimates for the regular extreme N-P components defined in the regular Hawking-Hartle tetrad is also proved. The hierarchy here is generalized in our second part (Ma in Uniform energy bound and Morawetz estimates for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole II: linearized gravity, 2017, arXiv:1708.07385) of this series to treat the extreme components of linearized gravity.

Local energy decay for scalar fields on time dependent non-trapping backgrounds

Author(s): Metcalfe, J; Sterbenz, J; Tataru, D | Abstract: We consider local energy decay estimates for solutions to scalar wave equations on nontrapping asymptotically flat space-times. Our goals are two-fold. First we consider the stationary case, where we can provide a full spectral characterization of local energy decay bounds; this characterization simplifies in the stationary symmetric case. Then we consider the almost stationary, almost symmetric case. There we establish two main results: The first is a “two point” local energy decay estimate which is valid for a general class of (non-symmetric) almost stationary wave equations which satisfy a certain nonresonance property at zero frequency. The second result, which also requires the almost symmetry condition, is to establish an exponential trichotomy in the energy space via finite dimensional time dependent stable and unstable sub-spaces, with an infinite dimensional complement on which solutions disperse via the usual local energy decay estimate.

ON LOCAL ENERGY DECAY ESTIMATE OF THE OSEEN SEMIGROUP IN TWO DIMENSIONS AND ITS APPLICATION

AbstractWe study the temporal decay estimate of the Oseen semigroup in a two-dimensional exterior domain. We establish the local energy decay estimate with a suitable dependence on the small translation speed, which is a significant improvement of Hishida’s result in 2016. As an application, we prove the $L^{q}$-$L^{r}$ estimates of the Oseen semigroup uniformly in the small translation speed.

A Vector Field Approach to Almost-Sharp Decay for the Wave Equation on Spherically Symmetric, Stationary Spacetimes

We present a new vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes. Specifically, we derive a new hierarchy of higher-order weighted energy estimates by employing appropriate commutator vector fields. In cases where an integrated local energy decay estimate holds, like in the case of sub-extremal Reissner–Nordstrom black holes, this hierarchy leads to almost-sharp global energy and pointwise time-decay estimates with decay rates that go beyond those obtained by the traditional vector field method. Our estimates play a fundamental role in our companion paper where precise late-time asymptotics are obtained for linear scalar fields on such backgrounds.

On a local energy decay estimate of solutions to the hyperbolic type Stokes equations

In this paper, we discuss a local energy decay estimate of solutions to the initial-boundary value problem for the hyperbolic type Stokes equations of incompressible fluid flow in an exterior domain and a perturbed half-space. The equations are linearized version of the hyperbolic Navier–Stokes equations introduced by Racke and Saal [15], which are obtained as a delayed case for the deformation tensor in the incompressible Navier–Stokes equations. Our proof of the local energy decay estimate is based on Dan and Shibata [2]. In [2], they treated the dissipative wave equations in an exterior domain and discussed the local energy decay estimate. Our approach uses the fact that applying the Helmholtz projection to the hyperbolic type Stokes equations, we obtain equations similar to the dissipative wave ones.

Global smooth solutions of 3-D null-form wave equations in exterior domains with Neumann boundary conditions

The paper is devoted to investigating long time behavior of smooth small data solutions to 3-D quasilinear wave equations outside of compact convex obstacles with Neumann boundary conditions. Concretely speaking, when the surface of a 3-D compact convex obstacle is smooth and the quasilinear wave equation fulfills the null condition, we prove that the smooth small data solution exists globally provided that the Neumann boundary condition on the exterior domain is given. One of the main ingredients in the current paper is the establishment of local energy decay estimates of the solution itself. As an application of the main result, the global stability to 3-D static compressible Chaplygin gases in exterior domain is shown under the initial irrotational perturbation with small amplitude.

Global existence for a coupled wave system related to the Strauss conjecture

A coupled system of semilinear wave equations is considered, and a small data global existence result related to the Strauss conjecture is proved. Previous results have shown that one of the powers may be reduced below the critical power for the Strauss conjecture provided the other power sufficiently exceeds such. The stability of such results under asymptotically flat perturbations of the space-time where an integrated local energy decay estimate is available is established.

Global existence for a coupled wave system related to the Strauss conjecture

A coupled system of semilinear wave equations is considered, and a small data global existence result related to the Strauss conjecture is proved. Previous results have shown that one of the powers may be reduced below the critical power for the Strauss conjecture provided the other power sufficiently exceeds such. The stability of such results under asymptotically flat perturbations of the space-time where an integrated local energy decay estimate is available is established.