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Abstract
In this note we prove global well-posedness for the defocusing, cubic nonlinear Schrödinger equation with initial data lying in a critical Sobolev space.
Highlights
It is straightforward to show that local well-posedness holds for (1.1) for any initial data u0 ∈ H 1/2(R3)
One cannot conclude directly from [CW90] that a uniform bound for u(t) H 1/2(R3) on the entire time of the existence of the solution to (1.1) implies that the solution is global
It is conjectured that (1.1) is globally well-posed and scattering for any u0 ∈ H 1/2(R3), without the a priori assumption of a universal bound on the H 1/2 norm of the solution u(t)
Summary
(1.1) is locally well-posed, and there exists some T (u0) > 0 such that (1.1) has a solution on [−T, T ]. This lemma allows us to make a Littlewood–Paley decomposition of unl, treat each Pjunl separately, and sum up. It implies that unl retains all the properties of a solution to the linear Schrodinger equation with initial data in a Besov space. Remark: Throughout this section we rely very heavily on the bilinear Strichartz estimate (2.5). Suppose that u is a solution to (1.1) satisfying (2.1). By the bilinear Strichartz estimate and the Sobolev embedding properties of Littlewood– Paley projections, 2j/2 (Pj−3≤·≤j+3u)(P≤j−3u) L1t L2x([−1,1]×R3). By Strichartz estimates, (2.3), Plancherel’s theorem, and the fractional product rule,
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