Finite time blow-up analysis for the generalized Proudman-Johnson model

TL;DR

This paper analyzes the finite time blow-up and global existence of solutions for the generalized Proudman-Johnson equation on the torus, revealing critical parameter regimes and self-similar blow-up behaviors.

Contribution

It provides a comprehensive study of blow-up phenomena and global solutions for the generalized Proudman-Johnson model, including critical parameter analysis and self-similarity.

Findings

Finite time blow-up for parameter a > 1 in the inviscid case.

Global existence for initial data when a < 1.

Self-similar blow-up in the inviscid Proudman-Johnson equation with Hölder continuous data.

Abstract

In this paper, we study the generalized Proudman-Johnson equation posed on the torus. In the critical regime where the parameter $a$ is close to and slightly greater than 1, we establish finite time blow-up of smooth solutions to the inviscid case. Moreover, we show that the blow-up is asymptotically self-similar for a class of smooth initial data. In contrast, when the parameter $a$ lies slightly below 1, we prove the global in time existence for the same initial data. In addition, we demonstrate that inviscid Proudman-Johnson equation with H\"{o}lder continuous data also develops a self-similar blow-up. Finally, for the viscous case with $a>1$, we prove that smooth initial data can still lead to finite time blow-up.