Well-Posedness for the Fifth Order KdV Equation

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 H-s for any s epsilon R. So we give initial data in H-s,H- (a) = H-s boolean AND H-a 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 H-s,H- a in the case s >= max{-1/4, -2a - 2}, -3/2 < a <= -1/4 and (s, a) not equal (-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).