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

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 نشر من قبل Tadahiro Oh
 تاريخ النشر 2018
  مجال البحث
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In this paper, we study the one-dimensional cubic nonlinear Schrodinger 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 $mathcal{F} L^p(mathbb{T})$, $1 leq p < infty$. 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 $1leq p leq frac32$.



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