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

Abstract

We derive a criterion for the breakdown of solutions to the Oldroyd-B model in R3 in the limit of zero Reynolds number (creeping flow). If the initial stress field is in the Sobolev space Hm(R3), 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 1 -norm of the stress tensor must diverge. This result is analogous to the celebrated Beale-Kato-Majda breakdown criterion for the inviscid Euler equations of incompressible fluids. AMS subject classifications. 35B35, 35Q35