A geometric view on the generalized Proudman-Johnson and $r$-Hunter-Saxton equations

TL;DR

This paper reveals a geometric interpretation of certain PDEs as geodesic flows on diffeomorphism groups, providing explicit solutions and insights into their behavior, unifying two families of equations.

Contribution

It establishes a new geometric framework for the Proudman--Johnson and $r$-Hunter--Saxton equations, showing their equivalence and analyzing their solutions using diffeomorphism group isometries.

Findings

Equivalence of Proudman--Johnson and $r$-Hunter--Saxton equations for certain parameters.

Explicit solutions for the non-periodic case.

Clarification of the limiting behavior of these equations.

Abstract

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.