# On the hierarchies of higher order mKdV and KdV equations

Open Mathematics (2010)

• Volume: 8, Issue: 3, page 500-536
• ISSN: 2391-5455

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## Abstract

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The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces ${\stackrel{^}{H}}_{s}^{r}\left(ℝ\right)$ defined by the norm ${∥{v}_{0}∥}_{{\stackrel{^}{H}}_{s}^{r}\left(ℝ\right)}:={∥{〈\xi 〉}^{s}\stackrel{^}{{v}_{0}}∥}_{{L}_{\xi }^{{r}^{\text{'}}}},〈\xi 〉={\left(1+{\xi }^{2}\right)}^{\frac{1}{2}},\frac{1}{r}+\frac{1}{{r}^{\text{'}}}=1$ . Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ $\frac{2j-1}{2{r}^{\text{'}}}$ . 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 < $\frac{2j-1}{2{r}^{\text{'}}}$ . 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 ${\stackrel{^}{H}}_{s}^{r}\left(ℝ\right)$ 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 ${\stackrel{^}{H}}_{s}^{r}\left(ℝ\right)$ , if 1 < r ≤ $\frac{2j}{2j-1}$ and $s>j-\frac{3}{2}-\frac{1}{2j}+\frac{2j-1}{2{r}^{\text{'}}}$ . For KdV itself the lower bound on s is pushed further down to $s>max\left(-\frac{1}{2}-\frac{1}{2{r}^{\text{'}}}-\frac{1}{4}-\frac{11}{8{r}^{\text{'}}}\right)$ , where r ∈ (1,2). These results rely on the contraction mapping principle, and the flow map is real analytic.

## How to cite

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Axel Grünrock. "On the hierarchies of higher order mKdV and KdV equations." Open Mathematics 8.3 (2010): 500-536. <http://eudml.org/doc/269252>.

@article{AxelGrünrock2010,
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 $\left\Vert \{v\_0 \} \right\Vert \_\{\hat\{H\}\_s^r \left( \mathbb \{R\} \right)\} : = \left\Vert \{\left\langle \xi \right\rangle ^s \widehat\{v\_0 \}\} \right\Vert \_\{L\_\xi ^\{r^\{\prime \}\} \} , \left\langle \xi \right\rangle = \left( \{1 + \xi ^2 \} \right)^\{\frac\{1\}\{2\}\} , \frac\{1\}\{r\} + \frac\{1\}\{\{r^\{\prime \}\}\} = 1$ . Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ $\frac\{\{2j - 1\}\}\{\{2r^\{\prime \}\}\}$ . 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 < $\frac\{\{2j - 1\}\}\{\{2r^\{\prime \}\}\}$ . 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 $s > j - \frac\{3\}\{2\} - \frac\{1\}\{\{2j\}\} + \frac\{\{2j - 1\}\}\{\{2r^\{\prime \}\}\}$ . For KdV itself the lower bound on s is pushed further down to $s > max\left( \{ - \frac\{1\}\{2\} - \frac\{1\}\{\{2r^\{\prime \}\}\} - \frac\{1\}\{4\} - \frac\{\{11\}\}\{\{8r^\{\prime \}\}\}\} \right)$ , where r ∈ (1,2). These results rely on the contraction mapping principle, and the flow map is real analytic.},
author = {Axel Grünrock},
journal = {Open Mathematics},
keywords = {mKdV and KdV hierarchies; Cauchy problem; Local and global well-posedness; Generalized Fourier restriction norm method; local and global well-posedness; generalized Fourier restriction norm method},
language = {eng},
number = {3},
pages = {500-536},
title = {On the hierarchies of higher order mKdV and KdV equations},
url = {http://eudml.org/doc/269252},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Axel Grünrock
TI - On the hierarchies of higher order mKdV and KdV equations
JO - Open Mathematics
PY - 2010
VL - 8
IS - 3
SP - 500
EP - 536
AB - 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 $\left\Vert {v_0 } \right\Vert _{\hat{H}_s^r \left( \mathbb {R} \right)} : = \left\Vert {\left\langle \xi \right\rangle ^s \widehat{v_0 }} \right\Vert _{L_\xi ^{r^{\prime }} } , \left\langle \xi \right\rangle = \left( {1 + \xi ^2 } \right)^{\frac{1}{2}} , \frac{1}{r} + \frac{1}{{r^{\prime }}} = 1$ . Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ $\frac{{2j - 1}}{{2r^{\prime }}}$ . 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 < $\frac{{2j - 1}}{{2r^{\prime }}}$ . 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 $s > j - \frac{3}{2} - \frac{1}{{2j}} + \frac{{2j - 1}}{{2r^{\prime }}}$ . For KdV itself the lower bound on s is pushed further down to $s > max\left( { - \frac{1}{2} - \frac{1}{{2r^{\prime }}} - \frac{1}{4} - \frac{{11}}{{8r^{\prime }}}} \right)$ , where r ∈ (1,2). These results rely on the contraction mapping principle, and the flow map is real analytic.
LA - eng
KW - mKdV and KdV hierarchies; Cauchy problem; Local and global well-posedness; Generalized Fourier restriction norm method; local and global well-posedness; generalized Fourier restriction norm method
UR - http://eudml.org/doc/269252
ER -

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