We study the initial value problem (IVP) associated to some canonical dispersive equations. Our main concern is to establish the minimal regularity property required in the data which guarantees the local well-posedness of the problem. Measuring this regularity in the classical Sobolev spaces, we show ill-posedness results for Sobolev index above the value suggested by the scaling argument.
"On the ill-posedness of some canonical dispersive equations." Duke Math. J. 106 (3) 617 - 633, 15 February 2001. https://doi.org/10.1215/S0012-7094-01-10638-8