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

The solutions of the Cauchy problem of the KdV equation on a periodic domain T,u(t) + uu(x) + u(xxx) = 0, u(x, 0) = phi(x), x is an element of T, t is an element of R,possess neither the sharp Kato smoothing property,phi is an element of H-s(T) double right arrow partial derivative(s+1)(x) u is an element of L-x(infinity) (T, L-2(0,T)),nor the Kato smoothing property,phi is an element of H-s(T) double right arrow u is an element of L-2(0,T;Hs+1(T)).Considered in this article is the Cauchy problem of the following dispersive equations posed on the periodic domain T,u(t) + uu(x) + u(xxx) - g(x)(g(x)u)(xx) = 0, u(x,0) = phi(x), x is an element of T, t > 0, (1)where g is an element of C-infinity(T) is a real value function with the supportomega = {x is an element of T, g(x)not equal 0}.It is shown that(1) if omega not equal theta, 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.