The Cauchy problem for a Bardina-Oldroyd model to the incompressible viscoelastic flow

In this paper, we study the Cauchy problem for a regularized viscoelastic fluid model in space dimension two, the Bardina-Oldroyd model, which is inspired by the simplified Bardina model for the turbulent flows of fluids, introduced by Cao et al. (2006). In particular, we obtain the local existence of smooth solutions to this model via the contraction mapping principle. Furthermore, we prove the global existence of smooth solutions to this system.