Kato smoothing properties of a class of nonlinear dispersive wave equations on a periodic domain^*

¹ Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA.
² Department of Mathematics, University of California at Riverside, Riverside, CA 92521, USA.
³ Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221, USA.

^** Corresponding author: zhangb@ucmail.uc.edu

Received: 10 August 2020
Accepted: 23 January 2021

Abstract

The solutions of the Cauchy problem of the KdV equation on a periodic domain 𝕋,

possess neither the sharp Kato smoothing property,

nor the Kato smoothing property,

Considered in this article is the Cauchy problem of the following dispersive equations posed on the periodic domain 𝕋, (See Eq. (1) below)

where g ∈ C^∞(𝕋) is a real value function with the support

It is shown that

(1) if ω ≠ ∅, then the solutions of the Cauchy problem (1) possess the Kato smoothing property;

(2) if g is a nonzero constant function, then the solutions of the Cauchy problem (1) possess the sharp Kato smoothing property.