Well-posedness for KdV-type equations with quadratic nonlinearity
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2018
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| _version_ | 1866929495040065536 |
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| author | Hirayama, Hiroyuki Kinoshita, Shinya Okamoto, Mamoru |
| author_facet | Hirayama, Hiroyuki Kinoshita, Shinya Okamoto, Mamoru |
| contents | We consider the Cauchy problem of the KdV-type equation \[ \partial_t u + \frac{1}{3} \partial_x^3 u = c_1 u \partial_x^2u + c_2 (\partial_x u)^2, \quad u(0)=u_0. \] Pilod (2008) showed that the flow map of this Cauchy problem fails to be twice differentiable in the Sobolev space $H^s(\mathbb{R})$ for any $s \in \mathbb{R}$ if $c_1 \neq 0$. By using a gauge transformation, we point out that the contraction mapping theorem is applicable to the Cauchy problem if the initial data are in $H^2(\mathbb{R})$ with bounded primitives. Moreover, we prove that the Cauchy problem is locally well-posed in $H^1(\mathbb{R})$ with bounded primitives. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1812_10002 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Well-posedness for KdV-type equations with quadratic nonlinearity Hirayama, Hiroyuki Kinoshita, Shinya Okamoto, Mamoru Analysis of PDEs We consider the Cauchy problem of the KdV-type equation \[ \partial_t u + \frac{1}{3} \partial_x^3 u = c_1 u \partial_x^2u + c_2 (\partial_x u)^2, \quad u(0)=u_0. \] Pilod (2008) showed that the flow map of this Cauchy problem fails to be twice differentiable in the Sobolev space $H^s(\mathbb{R})$ for any $s \in \mathbb{R}$ if $c_1 \neq 0$. By using a gauge transformation, we point out that the contraction mapping theorem is applicable to the Cauchy problem if the initial data are in $H^2(\mathbb{R})$ with bounded primitives. Moreover, we prove that the Cauchy problem is locally well-posed in $H^1(\mathbb{R})$ with bounded primitives. |
| title | Well-posedness for KdV-type equations with quadratic nonlinearity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1812.10002 |