Polynomial Bound and Nonlinear Smoothing for the Benjamin-Ono Equation on the Circle

Abstract

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 $(1+t{)}^{3(s-\frac{1}{2})+\epsilon }$, $0<\epsilon \ll 1$. 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.