Non-existence of solutions for the periodic cubic NLS below $L^2$

preprint2016arXivOpen access

We prove non-existence of solutions for the cubic nonlinear Schrödinger equation (NLS) on the circle if initial data belong to $H^s(\mathbb{T}) \setminus L^2(\mathbb{T})$ for some $s \in (-\frac18, 0)$ . The proof is based on establishing an a priori bound on solutions to a renormalized cubic NLS in negative Sobolev spaces via the short time Fourier restriction norm method.

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Non-existence of solutions for the periodic cubic NLS below $L^2$

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Non-existence of solutions for the periodic cubic NLS below L2L^2L2

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Non-existence of solutions for the periodic cubic NLS below $L^2$