The blow‐up criteria of smooth solutions to the generalized and ideal incompressible viscoelastic flow

In this paper, we prove two blow‐up criteria of smooth solution: one for the generalized incompressible Oldroyd model with fractional Laplacian velocity dissipation (?Δ)^αu in the space and one for the inviscid Oldroyd model. Assume that (u(t,x),F(t,x)) is a smooth solution to the generalized Oldroyd model in 0,T); then, the solution (u(t,x),F(t,x)) does not develop singularity until t = T provided . For the ideal impressible viscoelastic flow, it is shown that the smooth solution (u,F) can be extended beyond T if , which is an improvement of the result given by Hu and Hynd (A blowup criterion for ideal viscoelastic flow, J. Math. Fluid Mech., 15(2013), 431–437). Copyright © 2015 John Wiley & Sons, Ltd.