The blowup criterion of a smooth solution to the incompressible viscoelastic flow

In this paper, we prove the blowup criterion of a smooth solution to the incompressible Oldroyd model. Assume ( u ( t , x ) , F ( t , x ) ) is a smooth solution in [ 0 , T ) . The solution ( u ( t , x ) , F ( t , x ) ) does not develop singularity until t = T provided ∇ F ∈ L 1 ( 0 , T ; B M O ) in the case n = 2 , and provided ∇ u ∈ L 1 ( 0 , T ; B M O ) , ∇ F ∈ L 2 ( 0 , T ; B M O ) in the case n = 3 .