A Geometric View on the Generalized Proudman–Johnson and r-Hunter–Saxton Equations

We show that two families of equations, the generalized inviscid Proudman–Johnson equation and the r-Hunter–Saxton equation (recently introduced by Cotter et al.), coincide for a certain range of parameters. This gives a new geometric interpretation of these Proudman–Johnson equations as geodesic equations of right invariant homogeneous W^1,r-Finsler metrics on the diffeomorphism group. Generalizing a construction of Lenells for the Hunter–Saxton equation, we analyze these equations using an isometry from the diffeomorphism group to an appropriate subset of real-valued functions. Thereby, we show that the periodic case is equivalent to the geodesic equations on the L^r-sphere in the space of functions, and the non-periodic case is equivalent to a geodesic flow on a flat space. This allows us to give explicit solutions to these equations in the non-periodic case, and answer several questions of Cotter et al. regarding their limiting behavior.