A Beale-Kato-Majda breakdown criterion for an Oldroyd-B fluid in the creeping flow regime

TL;DR

This paper establishes a Beale-Kato-Majda type criterion for the breakdown of solutions to the Oldroyd-B model in creeping flow, linking solution existence to the divergence of the stress tensor's $L^ {infty}$ norm over time.

Contribution

It extends the Beale-Kato-Majda criterion to the Oldroyd-B model in the zero Reynolds number regime, providing a new condition for solution breakdown.

Findings

Solution exists indefinitely if the stress tensor's $L^ {infty}$ norm remains integrable.

Breakdown occurs only if the integral of the stress tensor's $L^ {infty}$ norm diverges.

The criterion parallels the classical result for Euler equations in fluid dynamics.

Abstract

We derive a criterion for the breakdown of solutions to the Oldroyd-B model in $\R^3$ in the limit of zero Reynolds number (creeping flow). If the initial stress field is in the Sobolev space $H^m$, $m> 5/2$, then either a unique solution exists within this space indefinitely, or, at the time where the solution breaks down, the time integral of the $L^\infty$-norm of the stress tensor must diverge. This result is analogous to the celebrated Beale-Kato-Majda breakdown criterion for the inviscid Eluer equations of incompressible fluids.