On the hierarchies of higher order mKdV and KdV equations

Axel Grünrock

Axel Grünrock

Open Mathematics, Tome 8 (2010), p. 500-536 / Harvested from The Polish Digital Mathematics Library

Résumé

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces ${\hat{H}}_{s}^{r} (ℝ)$ defined by the norm ${∥v_{0}∥}_{{\hat{H}}_{s}^{r} (ℝ)} : = {∥{〈ξ〉}^{s} \hat{v_{0}}∥}_{L_{ξ}^{r^{'}}}, 〈ξ〉 = {(1 + ξ^{2})}^{\frac{1}{2}}, \frac{1}{r} + \frac{1}{r^{'}} = 1$ . Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ $\frac{2 j - 1}{2 r^{'}}$ . 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 < $\frac{2 j - 1}{2 r^{'}}$ . 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 ${\hat{H}}_{s}^{r} (ℝ)$ cannot be solved by Picard iteration, if r > $\frac{2 j}{2 j - 1}$ , 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 ${\hat{H}}_{s}^{r} (ℝ)$ , if 1 < r ≤ $\frac{2 j}{2 j - 1}$ and $s > j - \frac{3}{2} - \frac{1}{2 j} + \frac{2 j - 1}{2 r^{'}}$ . For KdV itself the lower bound on s is pushed further down to $s > m a x (- \frac{1}{2} - \frac{1}{2 r^{'}} - \frac{1}{4} - \frac{11}{8 r^{'}})$ , where r ∈ (1,2). These results rely on the contraction mapping principle, and the flow map is real analytic.

Publié le : 2010-01-01

Zbl 1205.35253

EUDML-ID : urn:eudml:doc:269252

@article{bwmeta1.element.doi-10_2478_s11533-010-0024-5,
     author = {Axel Gr\"unrock},
     title = {On the hierarchies of higher order mKdV and KdV equations},
     journal = {Open Mathematics},
     volume = {8},
     year = {2010},
     pages = {500-536},
     zbl = {1205.35253},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0024-5}
}

Axel Grünrock. On the hierarchies of higher order mKdV and KdV equations. Open Mathematics, Tome 8 (2010) pp. 500-536. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0024-5/

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