On the 1D Cubic Nonlinear Schrödinger Equation in an Almost Critical Space

Abstract

We consider the Cauchy problem for the one dimensional cubic nonlinear Schrödinger equation iut + uxx − |u| 2 u = 0. As the first step local well-posedness in the modulation space M2,p (2 ≤ p < ∞) is derived (see Theorem 1.4), which covers all the subcritical cases. Afterwards in order to approach the endpoint case, we will prove the almost global wellposedness in some Orlicz type space (see Theorem 1.8), which is a natural generalization of M2,p, and is almost critical from the viewpoint of scaling. The new ingredient is an endpoint version of the two dimensional restriction estimate (see Lemma 3.7).

On global well-posedness of the modified KdV equation in modulation spaces

“…Our argument is based on bilinear estimates; see Lemmas 2.2 and Corollary 1. It is worthwhile to mention that our approach works equally well for the cubic NLS and the derivative NLS, providing an alternative approach to the results in [11,12].…”

“…In [24], we also established the same global-in-time a priori bound for solutions to the cubic NLS (1.2). Combining this with the local well-posedness of (1.2) in M 2,p s (R) for s ≥ 0 and 2 ≤ p < ∞ by S. Guo [11], we proved global well-posedness of the cubic NLS (1.2) in almost critical modulation spaces 4 M 2,p s (R) for s ≥ 0 On the other hand, there is no known local well-posedness for the modified KdV equation (1.1) in the modulation space M 2,p s (R), which motivated us to prove the following local well-posedness result.…”

“…Then, the complex-valued mKdV (1.1) is locally well-posed in M 2,p s (R). In [11], S. Guo proved local well-posedness of the cubic NLS (1.2) in the modulation spaces M 2,p s (R) for s ≥ 0 and 2 ≤ p < ∞. The proof was based on the Fourier restriction norm method adapted to the modulation spaces, where an endpoint version of two-dimensional Fourier restriction estimate played a crucial role.…”

“…See (2.3) below. We, however, provide a different approach than [11,12]. Our argument is based on bilinear estimates; see Lemmas 2.2 and Corollary 1.…”

On smoothing estimates in modulation spaces and the nonlinear Schrödinger equation with slowly decaying initial data

“…These estimates were further applied to prove well-posedness for nonlinear equations. For context we refer to S. Guo's work [29], in which he proved local well-posedness of the NLS in modulation spaces M 2,p for 2 < p < ∞. Oh-Wang [45] globalized this using the complete integrability.…”