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

Sun, Shu-Ming; Yang, Xin; Zhang, Bing-Yu; Zhong, Ning

ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 12

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.

Classification : 35B65, 35Q53, 35K45
Keywords: Kato smoothing property, sharp Kato smoothing property, KdV equation, KdV-Burgers equation

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