Smoothing and growth bound of periodic generalized Korteweg–De Vries equation

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For generalized Korteweg–De Vries (KdV) models with polynomial nonlinearity, we establish a local smoothing property in [Formula: see text] for [Formula: see text]. Such smoothing effect persists globally, provided that the [Formula: see text] norm does not blow up in finite time. More specifically, we show that a translate of the nonlinear part of the solution gains [Formula: see text] derivatives for [Formula: see text]. Following a new simple method, which is of independent interest, we establish that, for [Formula: see text], [Formula: see text] norm of a solution grows at most by [Formula: see text] if [Formula: see text] norm is a priori controlled.