Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szeg\H{o} equation

TL;DR

This paper investigates the growth of Sobolev norms for solutions to a quadratic Szegő equation on the torus, establishing optimal bounds and demonstrating exponential growth for certain solutions.

Contribution

It introduces a flow on BMO and proves the optimal exponential growth rate of Sobolev norms for solutions of the quadratic Szegő equation.

Findings

Constructed a flow propagating $H^s$ regularity for $s>0$

Demonstrated solutions with exponential Sobolev norm growth for $s>1/2$

Proved the exponential growth rate is optimal

Abstract

In this paper, we study a quadratic equation on the one-dimensional torus : $$i \partial_t u = 2J\Pi(|u|^2)+\bar{J}u^2, \quad u(0, \cdot)=u_0,$$ where $J=\int_\mathbb{T}|u|^2u \in\mathbb{C}$ has constant modulus, and $\Pi$ is the Szeg\H{o} 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.