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

[摘要] 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)+epsilon), 0 < epsilon << 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. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-10-05

[关键词] Periodic Benjamin-Ono equation;Nonlinear smoothing;Polynomial-in-time bound