Smoothing and growth bound of periodic generalized Korteweg-de Vries equation

For generalized KdV models with polynomial nonlinearity, we establish nonlinear smoothing property in $H^s$ for $s>\frac{1}{2}$. Such smoothing effect persists globally, provided that the $H^1$ 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 $s>\frac{1}{2}$. Following a new simple method, which is of independent interest, we establish that, for $s>1$, $H^s$ norm of a solution grows at most by $\langle t\rangle^{s-1+}$ if $H^1$ norm is a priori controlled.