For generalized Korteweg-De Vries (KdV) models with polynomial nonlinearity, we establish a local smoothing property in Hs for s > 1 2. Such smoothing effect persists globally, provided that the H1 norm does not blow up in finite time. More specifically, we show that a translate of the nonlinear part of the solution gains min(2s-1, 1)-derivatives for s > 1 2. Following a new simple method, which is of independent interest, we establish that, for s > 1, Hs norm of a solution grows at most by ts-1+ if H1 norm is a priori controlled.
