Global Well-Posedness and Scattering for Mass-Critical, Defocusing, Infinite Dimensional Vector-Valued Resonant Nonlinear Schrödinger System

Abstract

In this article, we consider the infinite dimensional vector-valued resonant nonlinear Schrödinger system, which arises from the study of the asymptotic behavior of the defocusing nonlinear Schrödinger equation on "wave guide" manifolds like R 2 × T in [7]. We show global well-posedness and scattering for this system by long time Strichartz estimates and frequency localized interaction Morawetz estimates. As a by-product, our results make the arguments of scattering theory in [7] closed as crucial ingredients for compactness of the critical elements.The other is the frequency localized interaction Morawetz estimate Theorem 1.3 (Frequency localized interaction Morawetz estimate). Suppose ⃗ u(t, x) is a minimal mass blowup solution to (2.1) on [0, T ] with ∫ T 0 N(t) 3 dt = K. Then (1.4) j∈Z ∇ 1 2 P ≤ 10K ǫ 1 u j (t, x) 2 2 L 2 t,x ([0,T ]×R 2 ) ≲ o(K), where o(K) is a quantity such that o(K) K → 0 as K → ∞.

On scattering for the cubic defocusing nonlinear Schrödinger equation on the waveguide $\mathbb R^2 \times \mathbb T$

“…Approximation of profiles. In this subsection, by using the solution of the resonant Schrödinger system in [26] to approximate the nonlinear Schrödinger equation with initial data be the bubble in the linear profile decomposition, we can show the nonlinear profile have a bounded space-time norm. Lemma 3.11 (Large-scale profiles ).…”

“…By Hölder's inequality, Sobolev's inequality, together with the scattering of the cubic resonant Schrödinger system and persistence of regularity in [26], we have…”

“…Title Scattering for the cubic NLS on R 2 × T Theorem 1.2 (Scattering of the cubic resonant Schrödinger system, [26]). Let E > 0, for any initial data…”

On the decay property of the cubic fourth-order Schrödinger equation

“…For dispersive models rather than 4NLS on Euclidean spaces (with scattering behavior), similar method may be applied to obtain the nonlinear decay property (i.e. the nonlinear solutions enjoy the same pointwise decay property as the linear solutions), such as, higher order (more than four) NLS, 4NLS on waveguide manifolds (see [31] for a recent result), NLS on waveguide manifolds (see [11,15,32] for example), NLS with partial harmonic potentials (see [1,3,12]), resonant system (see [4,30]), nonlinear wave equations (see [29]), Klein-Gordon equation (see [29]). We did not list them explicitly.…”

Global well-posedness and scattering of the two dimensional cubic focusing nonlinear Schrödinger system

“…On the other hand, for the infinite coupled nonlinear Schrödinger system, that is N = ∞, the nonlinear Schrödinger system appears in the nonlinear approximate of the cubic focusing nonlinear Schrödinger equations on the cylinder R 2 × T in [9], where it is called the resonant nonlinear Schrödinger system therein. The equation in the defocusing case was studied in [55], a similar nonlinear Schrödinger system is derived also in the study of the nonlinear Schrödinger equation with partial harmonic potentials in [6]. We also refer to [25] for a different view point.…”