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 $M{H}^{\theta ,p}$ whose norm is given by the ${\ell }^p$-sum of modulated $H^{\theta }$-norms of a given function. In particular, when $\theta <-\frac{1}{2}$, we show that the space $M{H}^{\theta ,p}$ 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 $\mathcal{F}{L}^p(\mathbb{T})$ 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.

26 pages. To appear in J. Differential Equations

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Cited by in corpus (5)

• Sharp well-posedness for the cubic NLS and mKdV in $H^s(\mathbb{R})$
• A remark on norm inflation for nonlinear Schrödinger equations
• On global well-posedness of the modified KdV equation in modulation spaces
• Stochastic nonlinear Schrödinger equation with almost space-time white noise
• Complex-valued solutions of the mKdV equations in generalized Fourier-Lebesgue spaces