Local well-posedness of the higher-dimensional $b$-equation
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910859029118976 |
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| author | Valletta, Justin |
| author_facet | Valletta, Justin |
| contents | The higher-dimensional $b$-equation is a family of PDEs, introduced by Holm and Staley (2003), that describe the motion of shallow water waves in $n$-dimensions. It expresses the invariance of the Lie-transport of the momentum one-form density associated with the fluid in $b$-dimensions. The constant $b$ can also be viewed as a balance parameter between fluid convection and fluid stretching/expansion. In this article, we interpret this family of PDEs as the geodesic equation of a right-invariant affine connection on the diffeomorphism group of $\mathbb{R}^n$. Using this framework and the methods of Ebin and Marsden (1970), we show local well-posedness of the $b$-equation with a Fourier multiplier as the inertia operator. This is achieved by formulating the $b$-equation as a smooth ODE on a Hilbert manifold, applying Picard-Lindelöf, and transferring back to the smooth category by showing that there is no loss of spatial regularity during the time evolution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03551 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local well-posedness of the higher-dimensional $b$-equation Valletta, Justin Analysis of PDEs The higher-dimensional $b$-equation is a family of PDEs, introduced by Holm and Staley (2003), that describe the motion of shallow water waves in $n$-dimensions. It expresses the invariance of the Lie-transport of the momentum one-form density associated with the fluid in $b$-dimensions. The constant $b$ can also be viewed as a balance parameter between fluid convection and fluid stretching/expansion. In this article, we interpret this family of PDEs as the geodesic equation of a right-invariant affine connection on the diffeomorphism group of $\mathbb{R}^n$. Using this framework and the methods of Ebin and Marsden (1970), we show local well-posedness of the $b$-equation with a Fourier multiplier as the inertia operator. This is achieved by formulating the $b$-equation as a smooth ODE on a Hilbert manifold, applying Picard-Lindelöf, and transferring back to the smooth category by showing that there is no loss of spatial regularity during the time evolution. |
| title | Local well-posedness of the higher-dimensional $b$-equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.03551 |