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

Abstract

Considered in this work is the initial value problem (IVP) associated to a higher order water wave model \begin{equation*} \begin{cases} η_t+η_x-γ_1 η_{xxt}+γ_2η_{xxx}+δ_1 η_{xxxxt}+δ_2η_{xxxxx}+\frac{3}{2}ηη_x+γ(η^2)_{xxx}-\frac{7}{48}(η_x^2)_x-\frac{1}{8}(η^3)_x=0,\\ η(x,0) = η_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 $H^{s,p}(\mathbb{R})$, $1\le p<\infty$. 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 $p\ge 1$, and in $H^{s,p}(\mathbb{R})$ whenever $p\in [1,\infty )$ and $s\ge \max \left\{\frac{1}{p}+\frac{1}{2},1\right\}$. 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 $T\gg 1$ if $1\le \frac{3}{2}-\frac{1}{p}<s<2$ or if $(s,p)\in [2,\infty ]\times [2,\infty ]$.

35 pages