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 b H s (R) defined by the norm ‖v0‖ b Hr s (R) := ‖〈ξ〉 b v0‖Lr′ ξ , 〈ξ〉 = (1 + ξ) 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 C0uniform sense, if s < 2j−1 2r . 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(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 b H s (R) cannot be solved by Picard iteration, if r > 2j 2j−1 , independent of the size of s ∈ R. Especially for j ≥ 2 we have C2-ill-posedness in H(R). With similar arguments as used before in the mKdV context it is shown that this problem is locally well-posed in b H 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 ∈ (1, 2). These results rely on the contraction mapping principle, and the flow map is real analytic.