On the ill-posedness of the cubic nonlinear Schrödinger equation on the circle

Abstract

In this note, we consider the ill-posedness issue for the cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, adapting the argument by Christ-Colliander-Tao [14] to the periodic setting, we exhibit a norm inflation phenomenon for both the usual cubic NLS and the Wick ordered cubic NLS for $s\le s_{\text{crit}}:=-\frac{1}{2}$. We also discuss norm inflation phenomena for general cubic fractional NLS on the circle.

30 pages. The critical case is replaced with a new argument. To appear in An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.)

References in corpus (4)

• Low regularity local well-posedness of the Derivative Nonlinear Schrödinger Equation with periodic initial data
• Instability of the periodic nonlinear Schrodinger equation
• Norm-inflation for periodic NLS equations in negative Sobolev spaces
• Non-existence of solutions for the periodic cubic NLS below $L^2$

Cited by in corpus (4)

• Solving the 4NLS with white noise initial data
• Norm inflation for the generalized Boussinesq and Kawahara equations
• A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces
• Non-existence of solutions for the periodic cubic NLS below $L^2$