The Role of Initial Curvature in Solutions to the Generalized Inviscid Proudman-Johnson Equation

TL;DR

This paper investigates how initial curvature affects solutions to the generalized inviscid Proudman-Johnson equation, extending previous results to broader initial data classes with arbitrary local curvature.

Contribution

It extends the analysis of solutions to the equation to include initial data with arbitrary local curvature, broadening the understanding of solution regularity.

Findings

Representation formulae for solutions are derived.

Regularity results are extended to wider initial data classes.

The influence of initial curvature on solution behavior is clarified.

Abstract

In [20], we derived representation formulae for spatially periodic solutions to the generalized, inviscid Proudman-Johnson equation and studied their regularity for several classes of initial data. The purpose of this paper is to extend these results to larger classes of functions including those having arbitrary local curvature near particular points in the domain.