Well-posedness for a higher order water wave model on modulation spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910594658992128 |
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| author | Carvajal, Xavier Panthee, Mahendra |
| author_facet | Carvajal, Xavier Panthee, Mahendra |
| contents | 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 $H^{s,p}(\mathbb{R})$, $1\leq p<\infty$. 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 $(s,p)\in [2, \infty]\times [2, \infty]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00467 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Well-posedness for a higher order water wave model on modulation spaces Carvajal, Xavier Panthee, Mahendra Analysis of PDEs 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 $H^{s,p}(\mathbb{R})$, $1\leq p<\infty$. 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 $(s,p)\in [2, \infty]\times [2, \infty]$. |
| title | Well-posedness for a higher order water wave model on modulation spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.00467 |