On the hierarchies of higher order mKdV and KdV equations

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces (H) over cap (r)(s)(R) defined by the norm parallel to v(0)parallel to(H) over cap (r)(s)(R) := parallel to <xi >(s)(v(0)) over cap parallel to L-xi r', <xi > = (1 + xi(2))(1/2), 1/r + 1/r' = 1. Local well-posedness for the jth equation is shown in the parameter range 2 >= r > 1, s >= 2j-1/2r'. The proof uses an appropriate variant of the Fourier restriction norm method. A counterexample is discussed to show that the Cauchy problem for equations of this type is in general ill-posed in the C-0-uniform sense, if s < 2j-1/2r'. 2 r 0. The results for r = 2 - so far in the literature only if j = 1(mKdV) or j = 2 - can be combined with the higher order conservation laws for the mKdV equation to obtain global well-posedness of the jth equation in H-s(R) for s >= j+1/2, if j is odd, and for s >= j/2, if j is even. - The Cauchy problem for the jth equation in the KdV hierarchy with data in <(H)over cap>(r)(s)(R) cannot be solved by Picard iteration, if r > 2j/2j-1, independent of the size of s is an element of R. Especially for j >= 2 we have C-2-ill-posedness in H-s(R). With similar arguments as used before in the mKdV context it is shown that this problem is locally well-posed in (H) over cap (r)(s)(R), if 1 < r <= 2j/2j-1 and s > j - 3/2 - 1/2j + 2j-1/2r'. For KdV itself the lower bound on s is pushed further down to s > max(-1/2 - 1/2r' - 1/4 - 11/8r'), where r is an element of (1,2). These results rely on the contraction mapping principle, and the flow map is real analytic.