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

TL;DR

This paper proves local well-posedness for the higher-dimensional $b$-equation, a family of PDEs modeling shallow water waves, by interpreting it as a geodesic equation on the diffeomorphism group and applying geometric analysis techniques.

Contribution

It establishes the local well-posedness of the higher-dimensional $b$-equation using geometric methods and the framework of affine connections on diffeomorphism groups.

Findings

Proves local existence and uniqueness of solutions.

Shows no loss of spatial regularity during evolution.

Extends well-posedness results to higher dimensions.

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-Lindel\"{o}f, and transferring back to the smooth category by showing that there is no loss of spatial regularity during the time evolution.