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arXiv · 2501.03551

Local well-posedness of the higher-dimensional $b$-equation

Justin Valletta
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Abstract

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.

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Justin Valletta. 2025-03-04. Local well-posedness of the higher-dimensional $b$-equation. https://arxiv.org/abs/2501.03551

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