Paper detail

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

In this paper, we first introduce a new function space $MH^{θ, p}$ whose norm is given by the $\ell^p$-sum of modulated $H^θ$-norms of a given function. In particular, when $θ< -\frac 12$, we show that the space $MH^{θ, p}$ agrees with the modulation space $M^{2, p}(\mathbb R)$ on the real line and the Fourier-Lebesgue space $\mathcal F L^{p}(\mathbb T)$ 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 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 $\mathbb R$ is globally well-posed in $M^{2, p}(\mathbb R)$ for any $p < \infty$, while the renormalized cubic NLS on $\mathbb T$ is globally well-posed in $\mathcal FL^p(\mathbb T)$ for any $p < \infty$. 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.