For initial data in Sobolev spaces $H^s(\mathbb{T})$, $\frac{1}{2}<s⩽1$, the solution to the Cauchy problem for the Benjamin-Ono equation on the circle is shown to grow at most polynomially in time at a rate ${(1+t)}^{3(s-\frac{1}{2})+ϵ}$, $0<ϵ\ll 1$. The key to establishing this result is the discovery of a nonlinear smoothing effect for the Benjamin-Ono equation, according to which the solution to the equation satisfied by a certain gauge transform, which is widely used in the well-posedness theory of the Cauchy problem, becomes smoother once its free solution is removed.

中文翻译：

---

圆上 Benjamin-Ono 方程的多项式边界和非线性平滑

对于 Sobolev 空间中的初始数据 $H^{秒}(\mathbb{吨})$, $\frac{1}{2}<秒⩽1$，圆上 Benjamin-Ono 方程的柯西问题的解显示为最多以多项式时间增长 ${(1+吨)}^{3(秒-\frac{1}{2})+\epsilon }$, $0<\epsilon \ll 1$. 建立这一结果的关键是发现了 Benjamin-Ono 方程的非线性平滑效应，根据该效应，方程的解满足某种规范变换，广泛应用于柯西问题的适定性理论, 一旦它的游离溶液被去除，就会变得更平滑。