Local well-posedness for dispersion generalized Benjamin-Ono equations in Fourier-Lebesgue spaces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916208043884544 |
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| author | Chen, Zijun |
| author_facet | Chen, Zijun |
| contents | We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation where $0<α\leq 1$ \begin{eqnarray*} \left\{ \begin{array}{l} \partial_t u+|\partial_x|^{1+α}\partial_x u+uu_x=0,\\ u(x,0)=u_0(x), \end{array} \right. \end{eqnarray*} is locally well-posed in the Fourier-Lebesgue space $\widehat{H}^{s}_{r}(\mathbb{R})$. This is proved via Picard iteration arguments using $X^{s,b}$-type space adapted to the Fourier-Lebesgue space, inspired by the work of Grünrock and Vega. Note that, previously, Molinet, Saut and Tzvetkov \cite{MST2001} proved that the solution map is not $C^2$ in $H^s$ for any $s$ if $0\leq α<1$. However, in the Fourier-Lebesgue space, we have a stronger smoothing effect to handle the $high\times low$ interactions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12353 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local well-posedness for dispersion generalized Benjamin-Ono equations in Fourier-Lebesgue spaces Chen, Zijun Analysis of PDEs 35E15, 35Q53 We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation where $0<α\leq 1$ \begin{eqnarray*} \left\{ \begin{array}{l} \partial_t u+|\partial_x|^{1+α}\partial_x u+uu_x=0,\\ u(x,0)=u_0(x), \end{array} \right. \end{eqnarray*} is locally well-posed in the Fourier-Lebesgue space $\widehat{H}^{s}_{r}(\mathbb{R})$. This is proved via Picard iteration arguments using $X^{s,b}$-type space adapted to the Fourier-Lebesgue space, inspired by the work of Grünrock and Vega. Note that, previously, Molinet, Saut and Tzvetkov \cite{MST2001} proved that the solution map is not $C^2$ in $H^s$ for any $s$ if $0\leq α<1$. However, in the Fourier-Lebesgue space, we have a stronger smoothing effect to handle the $high\times low$ interactions. |
| title | Local well-posedness for dispersion generalized Benjamin-Ono equations in Fourier-Lebesgue spaces |
| topic | Analysis of PDEs 35E15, 35Q53 |
| url | https://arxiv.org/abs/2403.12353 |