Finite-time singularity formation for scalar stretching equations

TL;DR

This paper demonstrates that a broad class of scalar stretching equations almost always develop finite-time singularities, even in energy-dissipating scenarios, with implications for physical fluid models.

Contribution

It provides a general proof of finite-time singularity formation for scalar equations involving linear operators, extending understanding beyond specific cases.

Findings

Finite-time singularities occur in most scalar stretching equations.

Singularities can form despite energy dissipation.

Applicable to models in inviscid and complex fluids.

Abstract

We consider equations of the type: \[\partial_t \omega = \omega R(\omega),\] for general linear operators $R$ in any spatial dimension. We prove that such equations almost always exhibit finite-time singularities for smooth and localized solutions. Singularities can even form in settings where solutions dissipate an energy. Such equations arise naturally as models in various physical settings such as inviscid and complex fluids.