Global in time Strichartz estimates for the fractional Schrodinger equations on asymptotically Euclidean manifolds

Van Duong Dinh (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

In this paper, we prove global in time Strichartz estimates for the fractional Schrödinger operators, namely $e^{−it\Lambda^\sigma_g}$ with $\sigma \in (0, \infty)\backslash \{1\}$ and $\Lambda_g := −\Delta_g$ where $\Delta_g$ 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 $(1 − f_0)(P)e^{−it\Lambda^\sigma_g}$ 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 a moderate trapping assumption, the high frequency part also satisfies the global in time Strichartz estimates outside a compact set. We next prove that the low frequency part $f_0 (P)e^{−it\Lambda^\sigma_g}$ satisfies global in time Strichartz estimates as on $\mathbb{R}^d$ of dimension $d \geq 3$ without using any geometric assumption on $g$. As a byproduct, we prove global in time Strichartz estimates for the fractional Schrödinger and wave equations on $(\mathbb{R}^d, g), d \geq 3$ under non-trapping condition.

Mots clés: Global in time Strichartz estimate; fractional Schr\"odinger equation; Littlewood-Paley decomposition; rescaled pseudo-differential operator; Isozaki-Kitada parametrix

Domaines: Equations aux dérivées partielles [math.AP]

Van Duong Dinh. Global in time Strichartz estimates for the fractional Schrodinger equations on asymptotically Euclidean manifolds. Journal of Functional Analysis, 2018, 275 (8), pp.1943-2014. ⟨hal-01522211⟩