[Paper Review] Strichartz estimates without loss outside two strictly convex obstacles

David Lafontaine|arXiv (Cornell University)|Sep 12, 2017

Advanced Mathematical Physics Problems15 references3 citations

TL;DR

This paper establishes global Strichartz estimates without loss for the Schrödinger equation outside two strictly convex obstacles in three dimensions, using microlocal analysis and stationary phase methods to handle trapped rays. The key result is $\|e^{-it\Delta_D}u_0\|_{L^p(\mathbb{R},L^q(\Omega))} \leq C\|u_0\|_{L^2}$ for admissible $(p,q)$ with $p>2$, $q\geq2$, $\frac{2}{p}+\frac{3}{q}=\frac{3}{2}$, despite the presence of a trapped periodic ray.

ABSTRACT

We prove global Strichartz estimates without loss outside two strictly convex obstacles, combining arguments from M.Ikawa (1982,1988) with more recent ones inspired by N.Burq, C.Guillarmou, and A. Hassell (2010) and O. Ivanovici (2010). This is to be contrasted with the loss in the smoothing estimate for the Schrödinger equation with Dirichlet boundary conditions in the presence of a trapped geodesic.

Motivation & Objective

• To establish global Strichartz estimates without loss for the Schrödinger equation in the presence of trapped geodesics due to two strictly convex obstacles.
• To extend the understanding of dispersive estimates beyond nontrapping geometries, particularly in the presence of boundary conditions.
• To overcome the logarithmic loss in the local smoothing estimate by proving Strichartz estimates up to times $h|\log h|$, compensating for the trapped ray.
• To demonstrate that Strichartz estimates can hold without loss even in a trapped setting with boundaries, a first such result for multiple convex obstacles.
• To provide a framework applicable in higher dimensions by leveraging dimension-independent components of the proof.

Proposed method

• Reduces the global Strichartz estimate to a local problem near the periodic trapped ray using frequency localization at scale $h^{-1}$.
• Applies microlocal techniques to restrict analysis to data microlocally supported near the trapped ray after logarithmic times.
• Constructs an approximate solution via the method of stationary phase for oscillatory integrals associated with reflected rays.
• Uses the stationary phase expansion with error control up to $h^2$ order, ensuring uniform bounds in the time interval $[0, \epsilon|\log h|]$.
• Combines estimates from small times ($t \leq t_0$) using free-space Schrödinger kernel decay $\sim (ht)^{-3/2}$ with large-time estimates via exponential decay.
• Employs the Christ-Kiselev lemma to avoid endpoint estimates, though the endpoint $(p,q)=(2,6)$ is not covered due to technical limitations.

Experimental results

Research questions

• RQ1 Can Strichartz estimates without loss be established in a trapped geometric setting with boundaries, such as two convex obstacles?
• RQ2 How does the presence of a periodic trapped ray affect the dispersive behavior of the Schrödinger flow in a bounded exterior domain?
• RQ3 Can the logarithmic loss in the local smoothing estimate be compensated by extending the time scale of Strichartz estimates to $h|\log h|$?
• RQ4 To what extent can the methods of [Ika88, Ika82] and [BGH10] be combined to handle trapped rays in multiple obstacle configurations?
• RQ5 Is the absence of loss in Strichartz estimates sensitive to the dimension, and can the 3D result be extended to higher dimensions?

Key findings

• Global Strichartz estimates without loss hold for the Schrödinger equation outside two strictly convex obstacles in $\mathbb{R}^3$ for all admissible $(p,q)$ with $p>2$, $q\geq2$, $\frac{2}{p}+\frac{3}{q}=\frac{3}{2}$.
• The estimate is uniform in time and does not require a loss in the Sobolev norm, even though a trapped periodic ray exists.
• The proof relies on constructing an approximate solution via stationary phase that captures the oscillatory behavior near the trapped ray for times up to $\epsilon|\log h|$.
• The size of the error terms in the stationary phase expansion is controlled by $h^{2}$, with additional decay from the geometry, ensuring $\sum_J |R_{\text{st.ph.}}^J| \lesssim h e^{-t/\epsilon}$.
• For small times $t \leq t_0$, the solution decays like $\sim (ht)^{-3/2}$, consistent with free-space dispersive estimates.
• The result is robust to dimension, as the core arguments extend to higher dimensions with minor modifications, though the paper focuses on $n=3$ due to foundational results in that setting.