2012 Volume 55 Issue 1 Pages 17-53
We consider the Cauchy problem of the fifth order KdV equation with low regularity data. We cannot apply the iteration argument to this problem when initial data is given in the Sobolev space Hs for any s ∈ R. So we give initial data in Hs,a = Hs ∩ Ha with a ≤ min{s, 0}. Then we recover more derivatives of the nonlinear term to be able to use the iteration method. Therefore we obtain the local well-posedness in Hs,a in the case s ≥ max{–1/4, –2a – 2}, –3/2 < a ≤ a –1/4 and (s,a) ≠ (–1/4, –7/8). Moreover, we obtain the ill-posedness in some sense when s < max{–1/4, –2a – 2}, a ≤ –3/2 or a > –1/4. The main tool is a variant of the Fourier restriction norm method, which is based on Kishimoto's work (2009).
