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

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.