On the hierarchies of higher order mKdV and KdV equations

Axel Grünrock

Open Mathematics (2010)

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

Abstract

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The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces H ^ s r ℝ defined by the norm v 0 H ^ s r ℝ : = ξ s v 0 ^ L ξ r ' , ξ = 1 + ξ 2 1 2 , 1 r + 1 r ' = 1 . Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ 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 < 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 ≥ 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 ^ s r ℝ cannot be solved by Picard iteration, if r > 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 H ^ s r ℝ , if 1 < r ≤ 2 j 2 j - 1 and s > j - 3 2 - 1 2 j + 2 j - 1 2 r ' . For KdV itself the lower bound on s is pushed further down to s > m a x - 1 2 - 1 2 r ' - 1 4 - 11 8 r ' , 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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