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) α u in the spaceR n , n = 2 , 3 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 ∇ u ∈ L 10 , T ;B ̇∞ , ∞ 0. For the ideal impressible viscoelastic flow, it is shown that the smooth solution ( u , F ) can be extended beyond T if ( ∇ u , ∇ F ) ∈ L 10 , T ;B ̇∞ , ∞ 0, 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.