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

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 $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> and show that this connection is Levi-Civita with respect to a right-invariant Riemannian metric only when $$b=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . 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.