A Beale-Kato-Majda 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∞-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. © 2008 International Press.