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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.

This paper is mainly concerned with the initial value problem (IVP) for the Korteweg-de Vries (KdV) equation { 8tu + 8;u + u8 x u = 0, (1.1) u(x, 0) = uo(x). X,tElR, The KdV equation, which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves [21], has been considered in different contexts, namely in its relation with the inverse scattering method, in plasma physics, and in algebraic geometry (see [24], and references therein). Our purpose is to study local and global well-posedness of the IVP (1.1) in classical Sobolev spaces H\lR). We shall say that the IVP (1.1) is locally (resp. globally) well-posed in the function space X if it induces a dynamical system on X by generating a continuous local (resp. global) flow. It was established in the works of Bona and Smith [3], Bona and Scott [2], Saut and Temam [30], and Kato [15] that the IVP (1.1) is locally (resp. globally) well-posed in H S with s > 3/2 (resp. s ~ 2). Roughly speaking, global well-posedness in H S depends on the available local theory and on the conservation laws satisfied by solutions of (1.1), namely: ~l(U) = ! udx, ~2(U) = ! u 2 dx , ~3(U) = ! ((8 x U)2-~ u 3