On Global-in-Time Strichartz Estimates for the Semiperiodic Schrödinger Equation

Abstract

We prove global-in-time Strichartz-type estimates for the Schrödinger equation on manifolds of the form R n × T d , where T d is a d-dimensional torus. Our results generalize and improve a global space-time estimate for the Schrödinger equation on R × T 2 due to Z. Hani and B. Pausader. As a consequence we prove global existence and scattering in H 1 2 for small initial data for the quintic NLS on R × T and the cubic NLS on R 2 × T.with optimal scaling for p near the Stein-Tomas endpoint in dimension n + d. Note however that it is relatively easy to prove such an estimate for large p.

Well-posedness for energy-critical nonlinear Schrödinger equation on waveguide manifold

“…For waveguide case, we recall the Strichartz estimate (local-in-time version) proved by Barron [1] as follows. The proof of Theorem 3.2 is based on the decoupling method established in Bourgain-Demeter [4].…”

“…Please see [19,20,22,23,29,34,35,36] for more information. The main purpose of this paper is to give a unified and simpler treatment of well-posedness results based on the Strichartz estimate established in Barron [1].…”

Long time dynamics for defocusing cubic NLS on three dimensional product space

“…Remark 1.3. It's expected that Theorem 1.1 holds for s > 1 2 since (1.1) is "H 1 2critical". Other methods or delicate techniques are required for one to obtain the sharp result.…”

“…the Strichartz estimate and the bilinear estimate in the waveguide setting is as same as in the tori setting. (See Barron [1] and Section 3 of this paper respectively.) Thus, in our previous result [12], we showed that we only need to care about the whole dimension of the waveguide, not the distribution of the Euclidean dimensions and the tori dimensions.…”

“…1) min(1, N −1 M )||P M Du|| L Then the contribution corresponding to the first term in (6.11) can be decomposed as (6.13)[0,T ]×M ∇Du•D (|u| 2 −|u 1 | 2 )∇u dxdt = K B [0,T ]×M ∇P 10B Du•D P K (|u| 2 −|u 1 | 2 )∇P B u dxdt,where for fixed K, B runs over some partition into cubes of size K. By orthogonality, this is bounded by(||DP K (|u| 2 − |u 1 | 2 )|| L o(1) M M o(1) min(1, N −1 M )||P M Du|| L…”