A geometric view on the generalized Proudman-Johnson and \(r\)-Hunter-Saxton equations

We show that two families of equations on the real line, 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 an appropriate diffeomorphism group on \(\mathbb{R}\). Generalizing a construction of Lenells for the Hunter--Saxton equation, we analyze the \(r\)-Hunter--Saxton equation 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 equation 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.