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

TL;DR

The paper analyzes the generalized Proudman–Johnson equation on the torus with a parameter $a$ and viscosity $ u$, focusing on the regime near $a=1$. It develops a dynamic rescaling framework and a weighted energy method to classify singularity formation, self-similar blow-up, and global existence across inviscid and viscous regimes. Key results include the construction of self-similar profiles for near-$a=1$ inviscid dynamics, finite-time blow-up for $a>1$ (and for viscous cases with small $|1-a|$), global existence for $a<1$, and Hölder-data blow-up at $a=1$, with precise energy estimates in weighted spaces. Together, these findings illuminate the delicate balance between convection and vortex-stretching in a one-parameter family of fluid models and provide rigorous criteria for singularity formation in the generalized Proudman–Johnston dynamics.

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 Hölder 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.

Theorems & Definitions (21)

• Definition 1.1: Weighted norms and spaces
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• proof : Proof of Lemma \ref{['le sinfx']}
• Lemma 2.3
• proof : Proof of Lemma \ref{['le estimates']}
• Lemma 2.4