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 ℱLp($${\cal F}{L^p}(\mathbb{T})$$ ℱ L p ( T ) ), 1 ≤ p < ∞. 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 ℱLp($${\cal F}{L^p}(\mathbb{T})$$ ℱ L p ( T ) ) for $$1 \leq p \leq {3 \over 2}$$ 1 ≤ p ≤ 3 2 .

Discrete Bilinear Operators and Commutators

“…In [13], the space 2 s is referred to as "discrete Sobolev space" (and denoted by h s in [13]). For general 1 ≤ p ≤ ∞, we have the following relation; f ∈ p s if and only if F −1 (f ) belongs to the so-called Fourier-Lebesgue space FL s.p (T d ) which plays an important role in the study of nonlinear PDEs; see, for example, [19,20]. It is easy to see that the dual of p s is p −s where p is the Hölder conjugate of p. Consider now an infinite tensor Θ : Z d × Z d × Z d → C; Θ will be identified with the collection of its elements Θ = {Θ(j, k, )} (j,k, )∈Z 3d .…”

Self-Similar Dynamics for the Modified Korteweg–de Vries Equation

“…We are then lead to the analysis of the full oscillatory integral in pt, ξq variables at critical regularity. As it is becoming clear in the recent years [7,18,26,27,28,29], the analogue of the Fourier restriction method in pt, ξq variables is the infinite normal form reduction (INFR). Let us briefly explain the idea behind this procedure.…”

“…The main difficulty that now arises is the derivation of multilinear bounds for every single term in the expansion, together with some decay in J to ensure summability. This has been for some time an unclear topic, especially in what concerns L 2 ξ -bounds: apart from some concrete cases [18,29], the general framework announced in [28] seems not to be completely correct (see Remark 4.5) and the validity of the approach described therein remains to be proved. In this work, we rigorously formalize the derivation of L 8 ξ a priori bounds through the INFR.…”

Uniqueness and non-uniqueness of the Gaussian free field evolution under the two-dimensional Wick ordered cubic wave equation

“…to study the dynamics with either random or deterministic initial data of low regularity. See [6,50,29,55]. Thanks to the L 2 -conservation, the equations (1.38) and (1.39) are equivalent, at least for smooth solutions, via the invertible gauge transform: u → e 2it ´|u| 2 dx u.…”