Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces

Abstract

In this paper, we first introduce a new function space MHθ,p whose norm is given by the ℓp-sum of modulated Hθ-norms of a given function. In particular, when θ<−12, we show that the space MHθ,p agrees with the modulation space M2,p(R) on the real line and the Fourier-Lebesgue space FLp(T) on the circle. We use this equivalence of the norms and the Galilean symmetry to adapt the conserved quantities constructed by Killip-Vişan-Zhang to the modulation space and Fourier-Lebesgue space setting. By applying the scaling symmetry, we then prove global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation (NLS) in almost critical spaces. More precisely, we show that the cubic NLS on R is globally well-posed in M2,p(R) for any p<∞, while the renormalized cubic NLS on T is globally well-posed in FLp(T) for any p<∞.In Appendix, we also establish analogous global-in-time bounds for the modified KdV equation (mKdV) in the modulation spaces on the real line and in the Fourier-Lebesgue spaces on the circle. An additional key ingredient of the proof in this case is a Galilean transform which converts the mKdV to the mKdV-NLS equation.