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))(u(t,x),F(t,x)) is a smooth solution in [0,T)[0,T). The solution (u(t,x),F(t,x))(u(t,x),F(t,x)) does not develop singularity until t=Tt=T provided ∇F∈L1(0,T;BMO)∇F∈L1(0,T;BMO) in the case n=2n=2, and provided ∇u∈L1(0,T;BMO)∇u∈L1(0,T;BMO), ∇F∈L2(0,T;BMO)∇F∈L2(0,T;BMO) in the case n=3n=3.