Sharp well-posedness and ill-posedness of the Cauchy problem for the higher-order KdV

In this paper, we investigate the Cauchy problem for the higher-order KdV-type equation \begin{eqnarray*} u_{t}+(-1)^{j+1}\partial_{x}^{2j+1}u

\frac{1}{2}\partial_{x}(u^{2}) = 0,j\in N^{+},x\in $T$ = [0,2\pi \lambda) \end{eqnarray*} with low regularity data and $λ \geq 1$ . Firstly, we show that the Cauchy problem for the periodic higher-order KdV equation is locally well-posed in $H^{s} (T)$ with $s \geq - j + \frac{1}{2}, j \geq 2.$ By using some new Strichartz estimate and some new function spaces, we also show that the Cauchy problem for the periodic higher-order KdV equation is ill-posed in $H^{s} (T)$ with $s < - j + \frac{1}{2}, j \geq 2$ in the sense that the solution map is $C^{3} .$ The result of this paper improves the result of \cite{H} with $j \geq 2$ .

In this paper, we investigate the Cauchy problem for the higher-order KdV-type equation \begin{eqnarray*} u_{t}+(-1)^{j+1}\partial_{x}^{2j+1}u

\frac{1}{2}\partial_{x}(u^{2}) = 0,j\in N^{+},x\in $T$ = [0,2\pi \lambda) \end{eqnarray*} with low regularity data and $λ \geq 1$ . Firstly, we show that the Cauchy problem for the periodic higher-order KdV equation is locally well-posed in $H^{s} (T)$ with $s \geq - j + \frac{1}{2}, j \geq 2.$ By using some new Strichartz estimate and some new function spaces, we also show that the Cauchy problem for the periodic higher-order KdV equation is ill-posed in $H^{s} (T)$ with $s < - j + \frac{1}{2}, j \geq 2$ in the sense that the solution map is $C^{3} .$ The result of this paper improves the result of \cite{H} with $j \geq 2$ .

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