2025/01/07 by Valletta, Justin
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2501.03551
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 ℝn. 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.