On the hierarchies of higher order mKdV and KdV equations

Abstract

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 $1<r\le \frac{2j}{2j-1}$ and $s > j - \frac32 - \frac{1}{2j} +\frac{2j-1}{2r'}$. 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.