On the hierarchies of higher order mKdV and KdV equations

Details On the hierarchies of higher order mKdV and KdV equations On the hierarchies of higher order mKdV and KdV equations

2009-09-16

arxiv.org/abs/0909.2971v2

Mathematics

Analysis of PDEs

36 pages, minor errors corrected in the second version

Scientific paper

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $\hat{H}^r_s(\R)$ defined by the norm $$\n{v_0}{\hat{H}^r_s(\R)} := \n{< \xi > ^s\hat{v_0}}{L^{r'}_{\xi}},\quad < \xi >=(1+\xi^2)^{\frac12}, \quad \frac{1}{r}+\frac{1}{r'}=1.$$ Local well-posedness for the $j$th equation is shown in the parameter range $2 \ge r >1$, $s \ge \frac{2j-1}{2r'}$. 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 $j$th equation in $H^s(\R)$ for $s\ge\frac{j+1}{2}$, if $j$ is odd, and for $s\ge\frac{j}{2}$, if $j$ is even. - The Cauchy problem for the $j$th equation in the KdV hierarchy with data in $\hat{H}^r_s(\R)$ cannot be solved by Picard iteration, if $r> \frac{2j}{2j-1}$, independent of the size of $s\in \R$. Especially for $j\ge 2$ 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 $\hat{H}^r_s(\R)$, if $1 j - \frac32 - \frac{1}{2j} +\frac{2j-1}{2r'}$. For KdV itself the lower bound on $s$ is pushed further down to $s>\max{(-\frac12-\frac{1}{2r'},-\frac14-\frac{11}{8r'})}$, where $r\in (1,2)$. These results rely on the contraction mapping principle, and the flow map is real analytic.

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