Well-posedness for a higher order water wave model on modulation spaces

Considered in this work is the initial value problem (IVP) associated to a higher order water wave model \begin{equation*} \begin{cases} \eta_t+\eta_x-\gamma_1 \eta_{xxt}+\gamma_2\eta_{xxx}+\delta_1 \eta_{xxxxt}+\delta_2\eta_{xxxxx}+\frac{3}{2}\eta \eta_x+\gamma (\eta^2)_{xxx}-\frac{7}{48}(\eta_x^2)_x-\frac{1}{8}(\eta^3)_x=0,\\ \eta(x,0) = \eta_0(x). \end{cases} \end{equation*} The main interest is in addressing the well-posedness issues of the IVP when the given initial data are considered in the modulation space $M_s^{2,p}(\mathbb{R})$ or the $L^p$-based Sobolev spaces $H^{s,p}(\mathbb{R})$, $1\leq p<\infty$. We derive some multilinear estimates in these spaces and prove that the above IVP is locally well-posed for data in $M_s^{2,p}(\mathbb{R})$ whenever $s>1$ and $p\geq 1$, and in $H^{s,p}(\mathbb{R})$ whenever $p\in [1,\infty)$ and $s\geq \max\left\{ \frac1{p}+\frac12, 1 \right\}$. We also use a combination of high-low frequency technique and an {\em a priori estimate}, and prove that the local solution with data in the modulation spaces $M_s^{2,p}(\mathbb{R})$ can be extended globally to the time interval $[0, T]$ for any given $T\gg1$ if $1\leq \frac32-\frac1p <s<2$ or if $(s,p)\in [2, \infty]\times [2, \infty]$.