On the Global Well-Posedness of the Inviscid Generalized\n Proudman-Johnson equation using flow map arguments

2019/02/11 by Florian Kogelbauer, Kogelbauer, Florian · 2 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1902.03787

openalex publication_date 2019/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We reformulate the Generalized Proudman--Johnson (GPJ) equation with\nparameter a in Lagrangian variables, where it takes the form of an\ninhomogeneous Liouville equation. This allows us to provide an explicitformula\nfor the flow map, up to the solution of an ODE. Depending on the parameter a,\nwe prove new criteria for global existence or formation of a finite-time\nsingularity. In particular, we show that there exist smooth initial data which\nbecome singular in finite time for a > 1. Also, we give a physical derivation\nof the GPJ equation for general parameter values of a.\n

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