On the Global Well-Posedness of the Inviscid Generalized Proudman-Johnson equation using flow map arguments

Abstract

We reformulate the Generalized Proudman--Johnson (GPJ) equation with parameter a in Lagrangian variables, where it takes the form of an inhomogeneous Liouville equation. This allows us to provide an explicitformula for the flow map, up to the solution of an ODE. Depending on the parameter a, we prove new criteria for global existence or formation of a finite-time singularity. In particular, we show that there exist smooth initial data which become singular in finite time for a > 1. Also, we give a physical derivation of the GPJ equation for general parameter values of a.