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 Rⁿ, 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 ϵ L¹ (0, T; B⁰ ∞,∞). For the ideal impressible viscoelastic flow, it is shown that the smooth solution (u, F) can be extended beyond T if (▿ u, ▿F) ϵ L¹ (0, T; B⁰ ∞,∞), 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). [ABSTRACT FROM AUTHOR]