Well-posedness and ill-posedness of KdV equation with higher dispersion

First, the Cauchy problem for KdV equation with 2 n + 1 order dispersion is studied, and the local well-posedness result for the initial data in Sobolev spaces H s ( R ) with s > − n + 1 4 is established via the Fourier restriction norm method. Second, we prove that the KdV equation with 2 n + 1 order dispersion is ill-posed for the initial data in H s ( R ) with s < − n + 1 4 , n ⩾ 2 , n ∈ N + if the flow map is C 2 differentiable at zero form H ˙ s ( R ) to C ( [ 0 , T ] ; H ˙ s ( R ) ) . Finally, we obtain the sharp regularity requirement for the KdV equation with 2 n + 1 order dispersion s > − n + 1 4 .