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

Abstract

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 Hs(R) with s>−n+14 is established via the Fourier restriction norm method. Second, we prove that the KdV equation with 2n+1 order dispersion is ill-posed for the initial data in Hs(R) with s<−n+14, n⩾2, n∈N+ if the flow map is C2 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+14.