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 \[ i ∂ t u = 2 J Π ( | u | 2 ) + J ¯ u 2 , u ( 0 , ⋅ ) = u 0 , i \partial _t u = 2J\Pi (|u|^2)+\bar {J}u^2, \quad u(0, \cdot )=u_0, \] where J = ∫ T | u | 2 u ∈ C J=\int _\mathbb {T}|u|^2u \in \mathbb {C} has constant modulus, and Π \Pi is the Szegő projector onto functions with nonnegative frequencies. Thanks to a Lax pair structure, we construct a flow on B M O ( T ) ∩ I m Π BMO(\mathbb {T})\cap \mathrm {Im}\Pi which propagates H s H^s regularity for any s > 0 s>0 , whereas the energy level corresponds to s = 1 / 2 s=1/2 . Then, for each s > 1 / 2 s>1/2 , we exhibit solutions whose H s H^s norm goes to + ∞ +\infty exponentially fast, and we show that this growth is optimal.