On the Cauchy problems for certain regularized models to the incompressible viscoelastic flow

In this paper, we study the Cauchy problems for certain regularized models to the incompressible viscoelastic flow in n space dimensions with n =2,3. Firstly, we establish a regularity condition for the solution under ∇ u ∈ L 1 ( 0 , T ; L ∞ ( R n ) ) . Furthermore, we obtain a regularity condition to the smooth solution for the inviscid regularized models in two space dimensions. Finally, we prove a global existence result of classical solutions for a three‐dimensional incompressible Oldroyd‐ α model with fractional diffusion.