Blowup of Solutions to a Damped Euler Equation with Homogeneous Three-Point Boundary Condition

TL;DR

This paper investigates conditions under which solutions to a damped, generalized inviscid Proudman-Johnson equation with three-point boundary conditions develop finite-time singularities, extending prior work on the undamped case.

Contribution

It introduces conditions on initial data that lead to finite-time blowup in solutions of the damped equation with specific boundary conditions.

Findings

Finite-time blowup can occur with bounded damping.

Finite-time blowup can occur with unbounded damping.

Derived explicit conditions on initial data for singularity formation.

Abstract

It has been established that solutions to the inviscid Proudman-Johnson equation subject to a homogeneous three-point boundary condition can develop singularities in finite time. In this paper, we consider the possibility of singularity formation in solutions of the generalized, inviscid Proudman-Johnson equation with damping subject to the same homogeneous three-point boundary condition. In particular, we derive conditions the initial data must satisfy in order for solutions to blowup in finite time with either bounded or unbounded smooth damping term.