Normal form approach to global well-posedness of the quadratic derivative nonlinear Schrödinger equation on the circle

Chung, Jaywan; Guo, Zihua; Kwon, Soonsik; Oh, Tadahiro

doi:10.1016/j.anihpc.2016.10.003

Chung, Jaywan ; Guo, Zihua ; Kwon, Soonsik ; Oh, Tadahiro

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1273-1297.

Résumé

We consider the quadratic derivative nonlinear Schrödinger equation (dNLS) on the circle. In particular, we develop an infinite iteration scheme of normal form reductions for dNLS. By combining this normal form procedure with the Cole–Hopf transformation, we prove unconditional global well-posedness in $L^{2} (T)$ , and more generally in certain Fourier–Lebesgue spaces $F L^{s, p} (T)$ , under the mean-zero and smallness assumptions. As a byproduct, we construct an infinite sequence of quantities that are invariant under the dynamics. We also show the necessity of the smallness assumption by explicitly constructing a finite time blowup solution with non-small mean-zero initial data.

MR Zbl

DOI : 10.1016/j.anihpc.2016.10.003

Classification : 35Q55
Mots clés : Quadratic derivative nonlinear Schrödinger equation, Normal form, Cole–Hopf transform, Fourier–Lebesgue space, Well-posedness, Finite time blowup

@article{AIHPC_2017__34_5_1273_0,
     author = {Chung, Jaywan and Guo, Zihua and Kwon, Soonsik and Oh, Tadahiro},
     title = {Normal form approach to global well-posedness of the quadratic derivative nonlinear {Schr\"odinger} equation on the circle},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1273--1297},
     publisher = {Elsevier},
     volume = {34},
     number = {5},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.10.003},
     mrnumber = {3742524},
     zbl = {1386.35376},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.10.003/}
}

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Chung, Jaywan; Guo, Zihua; Kwon, Soonsik; Oh, Tadahiro. Normal form approach to global well-posedness of the quadratic derivative nonlinear Schrödinger equation on the circle. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1273-1297. doi : 10.1016/j.anihpc.2016.10.003. http://www.numdam.org/articles/10.1016/j.anihpc.2016.10.003/

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