Local energy decay in even dimensions for the wave equation with a time-periodic non-trapping metric and applications to Strichartz estimates

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$R_\chi(\theta)=\chi(\mathcal U(T, 0)-e^{-i\theta})^{-1}\chi$, where$\mathcal U(T, 0)$is the monodromy operator and$T$the period of$a(t,x)$, admits an holomorphic continuation to$\{\theta\in\mathbb{C}\ :\ \textrm{Im}(\theta) \geq 0\}$, for$n \geq 3$, odd, and to$\{ \theta\in\mathbb C\ :\ \textrm{Im}(\theta)\geq0,\ \theta\neq 2k\pi-i\mu,\ k\in\mathbb{Z},\ \mu\geq0\}$for$n \geq4$, even, and for$n \geq4$even$R_\chi(\theta)$is bounded in a neighborhood of$\theta=0$

Local energy decay for the wave equation with a time-periodic non-trapping metric and moving obstacle

Consider the mixed problem with Dirichelet condition associated to the wave equation \partial_t^2u-\Div_{x}(a(t,x)\nabla_{x}u)=0, where the scalar metric $a(t,x)$ is $T$-periodic in $t$ and uniformly equal to 1 outside a compact set in $x$, on a $T$-periodic domain. Let $\mathcal U(t, 0)$ be the associated propagator. Assuming that the perturbations are non-trapping, we prove the meromorphic continuation of the cut-off resolvent of the Floquet operator $\mathcal U(T, 0)$ and establish sufficient conditions for local energy decay.Comment: Corrections of some misprint

On the meromorphic continuation of the resolvent for the wave equation with time-periodic perturbation and applications

Consider the wave equation $\partial_t^2u-\Delta_xu+V(t,x)u=0$, where $x\in\R^n$ with $n\geq3$ and $V(t,x)$ is $T$-periodic in time and decays exponentially in space. Let $U(t,0)$ be the associated propagator and let $R(\theta)=e^{-D}(U(T,0)-e^{-i\theta})^{-1}e^{-D}$ be the resolvent of the Floquet operator $U(T,0)$ defined for \im(\theta)>BT with $B>0$ sufficiently large. We establish a meromorphic continuation of $R(\theta)$ from which we deduce the asymptotic expansion of $e^{-(D+\epsilon)}U(t,0)e^{-D}f$, where $f\in \dot{H}^1(\R^n)\times L^2(\R^n)$, as $t\to+\infty$ with a remainder term whose energy decays exponentially when $n$ is odd and a remainder term whose energy is bounded with respect to $t^l\log(t)^m$, with $l,m\in\mathbb Z$, when $n$ is even. Then, assuming that $R(\theta)$ has no poles lying in $\{\theta\in\C\ :\ \im(\theta)\geq0\}$and is bounded for$\theta\to0$, we obtain local energy decay as well as global Strichartz estimates for the solutions of$\partial_t^2u-\Delta_xu+V(t,x)u=F(t,x)$

Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics

Doi proved that the $L^2_t H^{1/2}_x$ local smoothing effect for Schr\"odinger equation on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and $L^1\to L^\infty$ dispersive estimates still hold without loss for $e^{it\Delta}$ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.Comment: 23 pages. Corrections in the proof of prop 3.

Uniform resolvent estimates for Schr\"odinger operator with an inverse-square potential

We study the uniform resolvent estimates for Schr\"odinger operator with a Hardy-type singular potential. Let $\mathcal{L}_V=-\Delta+V(x)$ where $\Delta$ is the usual Laplacian on $\mathbb{R}^n$ and $V(x)=V_0(\theta) r^{-2}$ where $r=|x|, \theta=x/|x|$ and $V_0(\theta)\in\mathcal{C}^1(\mathbb{S}^{n-1})$ is a real function such that the operator $-\Delta_\theta+V_0(\theta)+(n-2)^2/4$ is a strictly positive operator on $L^2(\mathbb{S}^{n-1})$. We prove some new uniform weighted resolvent estimates and also obtain some uniform Sobolev estimates associated with the operator $\mathcal{L}_V$.Comment: Comments are welcome.To appear in Journal of Functional Analysi