Year: 2026
Type: article
Abstract: The higher-dimensional b -equation is a family of PDEs, introduced by Holm and Staley (SIAM J Appl Dyn Syst 2(3):323–380, 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, where the constant b can be thought of 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 $$\mathbb {R}^n$$ R n and show that this connection is Levi-Civita with respect to a right-invariant Riemannian metric only when $$b=2$$ b = 2 . Using this framework and the methods of Ebin and Marsden (Ann Math (2):92:102–163, 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.
Source: Journal of Evolution Equations
Author: Justin Valletta
Institution: Wake Forest University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
Award: G8653663419
OpenAlex ID: W7130938630