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

### Zihua Guo

School of Mathematical Sciences, Monash University, VIC 3800, Australia, LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China### Soonsik Kwon

Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, 34141, Republic of Korea### Tadahiro Oh

School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom, The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom### Jaywan Chung

National Institute for Mathematical Sciences, 70, Yuseong-daero 1689 beon-gil, Yuseong-gu, Daejeon, 34047, Republic of Korea

## Abstract

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 $FL_{s,p}(T)$, 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.

## Cite this article

Zihua Guo, Soonsik Kwon, Tadahiro Oh, Jaywan Chung, Normal form approach to global well-posedness of the quadratic derivative nonlinear Schrödinger equation on the circle. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 5, pp. 1273–1297

DOI 10.1016/J.ANIHPC.2016.10.003