Local well-posedness for a generalized sixth-order Boussinesq equation
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909131157274624 |
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| author | Zhong, Long Li, Shenghao |
| author_facet | Zhong, Long Li, Shenghao |
| contents | A formally second order correct Boussinesq-type equation that describes unidirectional shallow water waves is derived,
$$u_{tt} - u_{xx} - u_{xxxx} - u_{xxxxxx} - (u^2)_{xx} - (u^2)_{xxxx} - (uu_{xx})_{xx} - (u^3)_{xx} = 0.$$
Such equation is analogous to original Boussinesq equation but with higher order approximation which may ensure a more accuracy description on a long time scale. Moreover, through a rigorous derivation from Boussiensq systems, it has redeemed all the non-linear terms neglected in the sixth order Boussinesq equation (SOBE),
$$u_{tt} - u_{xx} - u_{xxxx} - u_{xxxxxx} - (u^2)_{xx} = 0.$$
The Cauchy problem for this generalized SOBE is then considered under the Bourgain space, $X^{s,b}$, framework. The multi-linear estimates for $(u^2)_{xx}$, $(u^2)_{xxxx}$, $(uu_{xx})_{xx}$ and $(u^3)_{xx}$ are given, the local wellposedness of the gSOBE is established for $s>\frac{1}{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04295 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local well-posedness for a generalized sixth-order Boussinesq equation Zhong, Long Li, Shenghao Analysis of PDEs A formally second order correct Boussinesq-type equation that describes unidirectional shallow water waves is derived, $$u_{tt} - u_{xx} - u_{xxxx} - u_{xxxxxx} - (u^2)_{xx} - (u^2)_{xxxx} - (uu_{xx})_{xx} - (u^3)_{xx} = 0.$$ Such equation is analogous to original Boussinesq equation but with higher order approximation which may ensure a more accuracy description on a long time scale. Moreover, through a rigorous derivation from Boussiensq systems, it has redeemed all the non-linear terms neglected in the sixth order Boussinesq equation (SOBE), $$u_{tt} - u_{xx} - u_{xxxx} - u_{xxxxxx} - (u^2)_{xx} = 0.$$ The Cauchy problem for this generalized SOBE is then considered under the Bourgain space, $X^{s,b}$, framework. The multi-linear estimates for $(u^2)_{xx}$, $(u^2)_{xxxx}$, $(uu_{xx})_{xx}$ and $(u^3)_{xx}$ are given, the local wellposedness of the gSOBE is established for $s>\frac{1}{2}$. |
| title | Local well-posedness for a generalized sixth-order Boussinesq equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.04295 |