Polynomial bound and nonlinear smoothing for the Benjamin-Ono equation on the circle

Abstract

For initial data in Sobolev spaces H s ( T ) , 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 − 1 2 ) + ϵ , 0 < ϵ ≪ 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.