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

First, the Cauchy problem for KdV equation with 2n +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 4 that the KdV equation with 2n + 1 order dispersion is ill-posed for the initial data in H-s (R) with s < -n + 1/4, n >= 2, n is an element of 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 2n + 1 order dispersion s > -n + 1/4. (C) 2014 Elsevier Inc. All rights reserved.