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

Kupferman, Raz; Mangoubi, Claude; Titi, Edriss S.

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

Authors

Kupferman, Raz

Mangoubi, Claude

Titi, Edriss S.

Description

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.