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

## TL;DR

This paper analyzes the inviscid Generalized Proudman-Johnson equation using flow map methods, providing explicit solutions and criteria for global existence or finite-time singularity depending on the parameter a.

## Contribution

It introduces a flow map formulation of the GPJ equation, deriving explicit solutions and new conditions for global well-posedness or singularity formation based on the parameter a.

## Key findings

- Explicit flow map formula up to an ODE
- Criteria for global existence depending on a
- Existence of finite-time singularities for a > 1

## 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.