Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szegő equation

Abstract

In this paper, we study a quadratic equation on the one-dimensional torus : $$i{\partial }_{t}u=2J\Pi (|u{|}^2)+\bar{J}{u}^2,u(0,\cdot )=u_0,$$ where $J={\int }_{\mathbb{T}}|u{|}^{2}u\in \mathbb{C}$ has constant modulus, and $\Pi$ is the Szegő projector onto functions with nonnegative frequencies. Thanks to a Lax pair structure, we construct a flow on BMO$(\mathbb{T})\cap \mathrm{Im}\Pi$ which propagates $H^s$ regularity for any $s>0$, whereas the energy level corresponds to $s=1/2$. Then, for each $s>1/2$, we exhibit solutions whose $H^s$ norm goes to $+\infty$ exponentially fast, and we show that this growth is optimal.

19 pages