Abstract
The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $$ \hat H_s^r \left( \mathbb{R} \right) $$ defined by the norm $$ \left\| {v_0 } \right\|_{\hat H_s^r \left( \mathbb{R} \right)} : = \left\| {\left\langle \xi \right\rangle ^s \widehat{v_0 }} \right\|_{L_\xi ^{r'} } , \left\langle \xi \right\rangle = \left( {1 + \xi ^2 } \right)^{\frac{1} {2}} , \frac{1} {r} + \frac{1} {{r'}} = 1 $$.
Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ $$ \frac{{2j - 1}} {{2r'}} $$. 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 < $$ \frac{{2j - 1}} {{2r'}} $$. 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 s(ℝ) for s ≥ $$ \frac{{j + 1}} {2} $$, if j is odd, and for s ≥ $$ \frac{j} {2} $$, if j is even. — The Cauchy problem for the jth equation in the KdV hierarchy with data in $$ \hat H_s^r \left( \mathbb{R} \right) $$ cannot be solved by Picard iteration, if r > $$ \frac{{2j}} {{2j - 1}} $$, independent of the size of s ∈ 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 $$ \hat H_s^r \left( \mathbb{R} \right) $$, if 1 < r ≤ $$ \frac{{2j}} {{2j - 1}} $$ and $$ s > j - \frac{3} {2} - \frac{1} {{2j}} + \frac{{2j - 1}} {{2r'}} $$. For KdV itself the lower bound on s is pushed further down to $$ s > max\left( { - \frac{1} {2} - \frac{1} {{2r'}} - \frac{1} {4} - \frac{{11}} {{8r'}}} \right) $$, where r ∈ (1,2). These results rely on the contraction mapping principle, and the flow map is real analytic.
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