Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces

Abstract

In this paper, we study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then prove that this normal form equation is unconditionally globally well-posed in the Fourier-Lebesgue spaces $\mathcal{F}{L}^p(\mathbb{T})$, $1\le p<\infty$. By inverting the transformation, we conclude global well-posedness of the renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable sense. This approach also allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in $\mathcal{F}{L}^p(\mathbb{T})$ for $1\le p\le \frac{3}{2}$.

32 pages. To appear in J. Anal. Math. Corrected typos in the references

References in corpus (1)

• Low regularity local well-posedness of the Derivative Nonlinear Schrödinger Equation with periodic initial data

Cited by in corpus (5)

• Solving the 4NLS with white noise initial data
• Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces
• Unconditional uniqueness of solutions for nonlinear dispersive equations
• On global well-posedness of the modified KdV equation in modulation spaces
• Stochastic nonlinear Schrödinger equation with almost space-time white noise