Blow-up of Solutions to the Generalized Inviscid Proudman-Johnson Equation

TL;DR

This paper investigates the finite-time blow-up behavior of solutions to the generalized inviscid Proudman-Johnson equation across all parameter values using a direct analytical approach.

Contribution

It provides explicit solution representations and analyzes blow-up phenomena for the generalized Proudman-Johnson equation for any real parameter.

Findings

Finite-time blow-up occurs for certain parameter ranges.

Explicit solution formulas are derived.

Blow-up conditions are characterized.

Abstract

For arbitrary values of a parameter $\lambda\in R$, finite-time blow-up of solutions to the generalized, inviscid Proudman-Johnson equation is studied via a direct approach which involves the derivation of representation formulae for solutions to the problem.