The Generalized Proudman–Johnson Equation Revisited

Abstract

We demonstrate the existence of solutions to the inviscid generalized Proudman-Johnson equation for parameters a lying in the open interval (−5, −1) which develop singularities in finite time; moreover, we show that there are solutions which exist for all times if a = −1. Finally, a simple blow-up criterion for solutions arising from a special class of initial data is given. Mathematics Subject Classification (2000). Primary 35B67; Secondary 35B35.

Global and singular solutions to the generalized Proudman–Johnson equation

“…[14,36].) Ifū ∈ F * , then the corresponding solution to the periodic generalized ProudmanJohnson equation u(t, .)…”

“…(3) was first derived by Okamoto and Zhu [30] and further investigated in [27,6,14,35,36]. Remarkably, the parameter a interconnects several well-studied equations within the framework of the generalized Proudman-Johnson equation (3), (5).…”

On the global well-posedness of the inviscid generalized Proudman–Johnson equation using flow map arguments

“…In [12,13] it is first noted that u x along trajectories satisfies a Riccati differential equation, which is equivalent to a second-order ODE for which then, in turn, knowledge of a special solution implies a representation formula for the general solution. In [6,14] blow-up conditions for a = 1 and general a were given by a method based on the time evolution of suprema and infima. For the special case a = 1 (Proudman-Johnson equation), different, more specific blow-up criteria can be obtained, cf.…”

Global Existence for the Generalized Two-Component Hunter–Saxton System

“…Through this link, the family of systems (1.1) also bridges the rich theories for the Burgers equation [4] (α = −2) and the axisymmetric Euler flow in R d [35,41] if α = 2/(d − 1). We also remark that if one sets ρ = √ −1 u x and κ = −α, the system (1.1) decouples to give, once again, the generalized Proudman-Johnson equation [7,35,48] with parameter a = 2α − 1. Other important special cases of the generalized Hunter-Saxton system (1.1) include the inviscid Kármán-Batchelor flow [5,6,21] for α = −κ = 1, which admits global strong solutions, and the celebrated Constantin-Lax-Majda equation [15] with α = −κ = ∞, a one-dimensional model for three-dimensional vorticity dynamics, which has an abundance of solutions blow-up in finite time.…”

“…It turns out that if this choice is made for arbitrary α ∈ R, one arrives at the generalized Proudman-Johnson equation [14,35,36,38,40,48] with parameter a = α − 1. Through this link, the family of systems (1.1) also bridges the rich theories for the Burgers equation [4] (α = −2) and the axisymmetric Euler flow in R d [35,41] if α = 2/(d − 1).…”