Polynomial representations for initial-boundary value problems involving the inviscid Proudman-Johnson equation

The central aim of this paper is to show how two-point Hermite interpolation can be used to construct polynomial representations of solutions to some initial-boundary-value problems for the inviscid Proudman-Johnson equation. This classic equation of fluid dynamics can be regarded as first-order hyperbolic, and an important by-product of our analysis is an understanding of how Hermite interpolation can be utilized for such equations. Different types of boundary conditions may result in finite time blow-up and/or large time approach to the steady state depending on the value of a parameter appearing in the problem.