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