We obtain the Strichartz inequalities $$ \| u \|_{L^q_t L^r_x([0,1] \times M)} \leq C \| u(0) \|_{L^2(M)}$$ for any smooth $n$-dimensional Riemannian manifold $M$ which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and non-trapping, where $u$ is a solution to the Schrödinger equation $iu_t + {1/2} Δ_M u = 0$, and $2 < q, r \leq \infty$ are admissible Strichartz exponents ($\frac{2}{q} + \frac{n}{r} = \frac{n}{2}$). This corresponds with the estimates available for Euclidean space (except for the endpoint $(q,r) = (2, \frac{2n}{n-2})$ when $n > 2$). These estimates imply existence theorems for semi-linear Schrödinger equations on $M$, by adapting arguments from Cazenave and Weissler \cite{cwI} and Kato \cite{kato}.
This result improves on our previous result in \cite{HTW}, which was an $L^4_{t,x}$ Strichartz estimate in three dimensions. It is closely related to the results of Staffilani-Tataru, Burq, Tataru, and Robbiano-Zuily, who consider the case of asymptotically flat manifolds.
We prove global Strichartz estimates (with spectral cutoff on the low frequencies) for non trapping metric perturbations of the Schroedinger equation, posed on the Euclidean space.
Consider a conical singular space $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. We study the Klein-Gordon equations with inverse-square potentials in the space $X$, proving in particular global-in-time Strichartz estimates in this setting.
We prove the global-in-time Strichartz estimates for wave equations on the nontrapping asymptotically conic manifolds. We obtain estimates for the full set of wave admissible indices, including the endpoint. The key points are the properties of the microlocalized spectral measure of Laplacian on this setting showed in Hassell-Zhang and a Littlewood-Paley squarefunction estimate. As applications, we prove the global existence and scattering for a family of nonlinear wave equations on this setting.
We prove global-in-time Strichartz estimates without loss of derivatives for the solution of the Schroedinger equation on a class of non-trapping asymptotically conic manifolds. We obtain estimates for the full set of admissible indices, including the endpoint, in both the homogeneous and inhomogeneous cases. This result improves on the results in earlier papers of Hassell-Tao-Wunsch and Mizutani which were local in time.
Applying the spectral measure estimates obtained in the author's joint work with A. Hassell, we establish global-in-time Strichartz estimates without loss via truncated / microlocalized dispersive estimates as well as energy estimates.
We provide reversed Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds using cluster estimates for spectral projectors proved previously in such generality. As a consequence, we solve a problem left open in \cite{SSWZ} about the endpoint case for global well-posedness of nonlinear wave equations. We also provide estimates in this context for the maximal wave operator.
We obtain global Strichartz estimates for the solution $u$ of the wave equation $\partial_t^2 u-\Div_x(a(t,x)\nabla_xu)=0$ with time-periodic metric $a(t,x)$ equal to 1 outside a compact set with respect to $x$. We assume $a(t,x)$ is a non-trapping perturbation and moreover, we suppose that there are no resonances $z_j\in\mathbb{C}$ with $|z_j|\geq1$.
We prove global-in-time Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds. The key tools are the spectral measure estimates from \cite{CH2} and arguments borrowed from \cite{HZ, Zhang}. As an application, we prove the small data global existence for any power $p\in(1, 1+\frac{4}{n-1})$ for the shifted wave equation in this setting, involving nonlinearities of the form $\pm|u|^p$ or $\pm|u|^{p-1}u$, which answers partially an open question raised in \cite{SSW}.
We obtain an $L^4$ space-time Strichartz inequality for any smooth three-dimensional Riemannian manifold $(M,g)$ which is asymptotically conic at infinity and non-trapping, where $u$ is a solution to the Schrödinger equation $iu_t + {1/2} Δ_M u = 0$. The exponent $H^{1/4}(M)$ is sharp, by scaling considerations. In particular our result covers asymptotically flat non-trapping manifolds. Our argument is based on the interaction Morawetz inequality introduced by Colliander et al., interpreted here as a positive commutator inequality for the tensor product $U(t,z',z'') := u(t,z') u(t,z'')$ of the…
We consider an $n$-dimensional spherically symmetric, asymptotically Euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic. By separating variables and constructing a semiclassical parametrix for a time scale polynomially beyond Ehrenfest time, we show that solutions to the linear Schrödiner equation with initial conditions localized on a spherical harmonic satisfy Strichartz estimates with a loss depending only on the dimension $n$ and independent of the degeneracy. The Strichartz estimates are sharp up to an arbitrary $β>0$ loss. This is in contras…
We study the nonlinear Klein-Gordon equation on a product space $M=\R\times X$ with metric $\tilde{g}=dt^2-g$ where $g$ is the scattering metic on $X$. We establish the global-in-time Strichartz estimate for Klein-Gordon equation without loss of derivative by using the microlocalized spectral measure of Laplacian on scattering manifold showed in \cite{HZ} and a Littlewood-Paley squarefunction estimate proved in \cite{Zhang}. We prove the global existence and scattering for a family of nonlinear Klein-Gordon equations for small initial data with minimum regularity on this setting.
We prove global Strichartz inequalities for the Schrödinger equation on a large class of asymptotically conical manifolds. Letting $ P $ be the nonnegative Laplace operator and $ f_0 \in C_0^{\infty}({\mathbb R}) $ be a smooth cutoff equal to $1$ near zero, we show first that the low frequency part of any solution $ e^{-itP} u_0 $, i.e. $ f_0 (P) e^{-itP} u_0 $, enjoys the same global Strichartz estimates as on $ {\mathbb R}^n $ in dimension $ n \geq 3 $. We also show that the high energy part $ (1-f_0)(P) e^{-itP} u_0$ also satisfies global Strichartz estimates without loss of derivatives out…
We obtain local energy decay as well as global Strichartz estimates for the solutions $u$ of the wave equation $\partial_t^2 u-div_x(a(t,x)\nabla_xu)=0,\ t\in{\R},\ x\in{\R}^n,$ with time-periodic non-trapping metric $a(t,x)$ equal to $1$ outside a compact set with respect to $x$. We suppose that the cut-off resolvent $R_χ(θ)=χ(\mathcal U(T, 0)-e^{-iθ})^{-1}χ$, where $\mathcal U(T, 0)$ is the monodromy operator and $T$ the period of $a(t,x)$, admits an holomorphic continuation to $\{θ\in\mathbb{C}\ :\ \textrm{Im}(θ) \geq 0\}$, for $n \geq 3$ , odd, and to $\{ θ\in\mathbb C\ :\ \textrm{Im}(θ)\g…
The authors prove global Strichartz estimates for compact perturbations of the wave operator in odd dimensions when a non-trapping assumption is satisfied.
In this paper, we prove global in time Strichartz estimates for the fractional Schrödinger operators, namely $e^{-itΛ_g^σ}$ with $σ\in (0,\infty)\backslash \{1\}$ and $Λ_g:=\sqrt{-Δ_g}$ where $Δ_g$ is the Laplace-Beltrami operator on asymptotically Euclidean manifolds $(\mathbb{R}^d,g)$. Let $f_0\in C^\infty_0(\mathbb{R})$ be a smooth cutoff equal 1 near zero. We firstly show that the high frequency part $(1-f_0)(P)e^{-itΛ_g^σ}$ satisfies global in time Strichartz estimates as on $\mathbb{R}^d$ of dimension $d\geq 2$ inside a compact set under non-trapping condition. On the other hand, under t…
We show global-in-time Strichartz estimates for the isotropic Maxwell system with divergence free data. On the scalar permittivity and permeability we impose decay assumptions as $|x|\to\infty$ and a non-trapping condition. The proof is based on smoothing estimates in weighted $L^2$ spaces which follow from corresponding resolvent estimates for the underlying Helmholtz problem.
In this article we study global-in-time Strichartz estimates for the Schrödinger evolution corresponding to long-range perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. In the aforementioned paper, the third author proved the local smoothing estimates 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 compac…
We prove the (local in time) Strichartz estimates (for the full range of parameters given by the scaling unless the end point) for asymptotically flat and non trapping perturbations of the flat Laplacian in $\R^n$, $n\geq 2$. The main point of the proof, namely the dispersion estimate, is obtained in constructing a parametrix. The main tool for this construction is the use of the FBI transform.
We study local in time Strichartz estimates for the Schroedinger equation associated to long range perturbations of the flat Laplacian on the euclidean space. We prove that in such a geometric situation, outside of a large ball centered at the origin, the solutions of the Schroedinger equation enjoy the same Strichartz estimates as in the non perturbed situation. The proof is based on the Isozaki-Kitada parametrix construction. If in addition the metric is non trapping, we prove that the Strichartz estimates hold in the whole space.
We prove Strichartz estimates for Maxwell equations in media in the fully anisotropic case with Hölder-continuous coefficients. To this end, we use the FBI transform to conjugate the problem to phase space. After reducing to a scalar estimate by means of a matrix symmetrizer, we show oscillatory integral estimates for a variable-coefficient Fourier extension operator. The characteristic surface has conical singularities for any non-vanishing time frequency. Combined with energy estimates, we improve the local well-posedness for certain fully anisotropic quasilinear Maxwell equations.
In the present paper we consider Schrödinger equations with variable coefficients and potentials, where the principal part is a long-range perturbation of the flat Laplacian and potentials have at most linear growth at spatial infinity. We then prove local-in-time Strichartz estimates, outside a large compact set centered at origin, expect for the endpoint. Moreover we also prove global-in-space Strichartz estimates under the non-trapping condition on the Hamilton flow generated by the kinetic energy.
In this article, we shall construct the resolvent of Laplacian at high energies near the spectrum on non-product conic manifolds with a single cone tip. Microlocally, the resolvent kernel is the sum of b-pseudodifferential operators, scattering pseudodifferential operators and intersecting Legendrian distributions. As an application, we shall establish Strichartz estimates for Schrödinger equations on non-compact manifolds with multiple non-product conic singularities.
We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of earlier work of Metcalfe-Tataru, in order to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere.
In this paper we show a general Strichartz estimate for certain perturbed wave equation, and here we can drop the nontrapping hypothesis and handle trapping obstacles with some loss of derivatives for data in the local energy decay estimates. Then we give the obstacle version of sharp life span for semilinear wave equations when $n=3,p<p_c$, by using a real interpolation method, and by getting a corresponding finite time Strichartz estimates(see section 3). Finally, as an application of the general Strichartz estimates we have gained, we get the Strauss conjecture for semilinear wave equations…
We prove Strichartz estimates with a loss of derivatives for the Schrödinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz …
We consider the solution operator for the wave equation on the flat Euclidean cone over the circle of radius $ρ> 0$, the manifold $\mathbb{R}_+ \times \mathbb{R} / 2 πρ\mathbb{Z}$ equipped with the metric $\g(r,θ) = dr^2 + r^2 dθ^2$. Using explicit representations of the solution operator in regions related to flat wave propagation and diffraction by the cone point, we prove dispersive estimates and hence scale invariant Strichartz estimates for the wave equation on flat cones. We then show that this yields corresponding inequalities on wedge domains, polygons, and Euclidean surfaces with coni…
We prove global wellposedness in the energy space of the defocusing cubic nonlinear Schroedinger and Gross-Pitaevskii equations on the exterior of a non-trapping domain in dimension 3. The main ingredient is a Strichartz estimate obtained combining a semi-classical Strichartz estimate with a smoothing effect on exterior domains.
Consider the metric cone $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$ where the cross section $Y$ is a compact $(n-1)$-dimensional Riemannian manifold $(Y,h)$. Let $Δ_g$ be the Friedrich extension positive Laplacian on $X$ and let $Δ_h$ be the positive Laplacian on $Y$, and consider the operator $\LL_V=Δ_g+V_0 r^{-2}$ where $V_0\in\CC^\infty(Y)$ such that $Δ_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$. In this paper, we prove the global-in-time Strichartz estimates without loss for the wave equation associated with the operator $\LL_V$ which verifies\ci…
We establish Strichartz estimates for the Schrödinger equation on Riemannian manifolds $(Ω,\g)$ with boundary, for both the compact case and the case that $Ω$ is the exterior of a smooth, non-trapping obstacle in Euclidean space. The estimates for exterior domains are scale invariant; the range of Lebesgue exponents $(p,q)$ for which we obtain these estimates is smaller than the range known for Euclidean space, but includes the key $L^4_tL^\infty_x$ estimate, which we use to give a simple proof of well-posedness results for the energy critical Schrödinger equation in 3 dimensions. Our estimate…
This is the first in a series of papers in which we investigate the resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds with applications to the restriction theorem, spectral multiplier results and Strichartz estimates. In this first paper, we use semiclassical Lagrangian distributions and semiclassical intersecting Lagrangian distributions, along with Mazzeo-Melrose 0-calculus, to construct the high energy resolvent on general non- trapping asymptotically hyperbolic manifolds, generalizing the work due to Melrose, Sa Barreto and Vasy. We note that there is an in…
We prove Strichartz estimates for the Schroedinger equation with an electromagnetic potential, in dimension $n\geq3$. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition, we require repulsivity and a non trapping condition, which are expressed as smallness of suitable components of the potentials. However, the potentials themselves can be large, and we avoid completely any a priori spectral assumption on the operator. The proof is based on smoothing estimates and new Sobolev embeddings for spaces associated to magnetic potent…
We obtain KSS, Strichartz and certain weighted Strichartz estimate for the wave equation on $(\R^d, \mathfrak{g})$, $d \geq 3$, when metric $\mathfrak{g}$ is non-trapping and approaches the Euclidean metric like $ x ^{- ρ}$ with $ρ>0$. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for $ρ> 1$ and $d=3$. Also, we establish the Strauss conjecture when the metric is radial with $ρ>0$ for $d= 3$.
We consider the Laplacian $Δ$ on an asymptotically hyperbolic manifold $X$, as defined by Mazzeo and Melrose. We give pointwise bounds on the spectral measure for the operator $(Δ- n^2/4)_+^{1/2}$ on such manifolds, under the assumptions that $X$ is nontrapping and there is no resonance at the bottom of the spectrum. This uses the construction of the resolvent given by Mazzeo and Melrose (valid when the spectral parameter lies in a compact set), Melrose, Sá Barreto and Vasy (high energy estimates for a perturbation of the hyperbolic metric) and the present authors (see also the work by Yiran W…
In this article, we study the properties of the geodesic X-ray transform for asymptotically Euclidean or conic Riemannian metrics and show injectivity under non-trapping and no conjugate point assumptions. We also define a notion of lens data for such metrics and study the associated inverse problem.
Global in time dispersive estimates for the Schroedinger and wave evolutions are obtained on manifolds with conical ends whose Hamiltonian flow exhibits trapping. This paper deals with the case of initial data with fixed "nonzero angular momentum".
We study eigenvalues of non-self-adjoint Schrödinger operators on non-trapping asymptotically conic manifolds of dimension $n\ge 3$. Specifically, we are concerned with the following two types of estimates. The first one deals with Keller type bounds on individual eigenvalues of the Schrödinger operator with a complex potential in terms of the $L^p$-norm of the potential, while the second one is a Lieb-Thirring type bound controlling sums of powers of eigenvalues in terms of the $L^p$-norm of the potential. We extend the results of Frank (2011), Frank-Sabin (2017), and Frank-Simon (2017) on th…
We prove uniform Sobolev estimates $||u||_{L^{p'}} \leq C ||(Δ-α)u||_{L^{p}}$, where $p=2n/(n+2), p'=2n/(n-2)$, for the Laplacian $Δ$ on non-trapping asymptotically conic manifolds of dimension $n$. Here C is independent of $α$ which ranges over all complex numbers. This generalizes to non-constant coefficient Laplacians a result of Kenig-Ruiz-Sogge.
Global in time dispersive estimates for the Schroedinger and wave evolutions are obtained on manifolds with conical ends whose Hamiltonian flow exhibits trapping. This paper deals with the case of initial data with "zero angular momentum".
We consider manifolds with conic singularites that are isometric to $\mathbb{R}^{n}$ outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via a doubling process. The proof of the resolvent estimate relies on the propagation of singularities theorems of Melrose and the second author to establish a "very weak" Huygens' principle, which may be of independent interest. As applications of the estimate, we obtain…
Let g be a scattering metric on a compact manifold X with boundary, i.e., a smooth metric giving the interior of X the structure of a complete Riemannian manifold with asymptotically conic ends. An example is any compactly supported perturbation of the standard metric on Euclidean space. Consider the operator $H = \half Δ+ V$, where $Δ$ is the positive Laplacian with respect to g and V is a smooth real-valued function on X vanishing to second order at the boundary. Assuming that g is non-trapping, we construct a global parametrix for the kernel of the Schroedinger propagator $U(t) = e^{-itH}$ …
In this paper we consider the defocusing energy critical wave equation with a trapping potential in dimension $3$. We prove that the set of initial data for which solutions scatter to an unstable excited state $(ϕ, 0)$ forms a finite co-dimensional path connected $C^1$ manifold in the energy space. This manifold is a global and unique center-stable manifold associated with $(ϕ,0)$. It is constructed in a first step locally around any solution scattering to $ϕ$, which might be very far away from $ϕ$ in the $\dot{H}^1\times L^2(\mathbb{R}^3)$ norm. In a second crucial step a no-return property i…
We consider abstract non-negative self-adjoint operators on $L^2(X)$ which satisfy the finite speed propagation property for the corresponding wave equation. For such operators we introduce a restriction type condition which in the case of the standard Laplace operator is equivalent to $(p,2)$ restriction estimate of Stein and Tomas. Next we show that in the considered abstract setting our restriction type condition implies sharp spectral multipliers and endpoint estimates for the Bochner-Riesz summability. We also observe that this restriction estimate holds for operators satisfying dispersiv…
We investigate quantitative (or effective) versions of the limiting absorption principle, for the Schrödinger operator on asymptotically conic manifolds with short-range potentials, and in particular consider estimates of the form $$ \| R(λ+i\eps) f \|_{H^{0,-1/2-σ}} \leq C(λ, H) \| f \|_{H^{0,1/2+σ}}.$$ We are particularly interested in the exact nature of the dependence of the constants $C(λ,H)$ on both $λ$ and $H$. It turns out that the answer to this question is quite subtle, with distinctions being made between low energies $λ\ll 1$, medium energies $λ\sim 1$, and large energies $λ\gg 1$,…
