On the well-posedness of the ideal incompressible viscoelastic flow in the critical Besov spaces

In this paper, we study the viscoelastic fluid system of the Oldroyd model. By means of the Fourier frequency localization and Bony paraproduct decomposition, we establish the local well-posedness for the incompressible Oldroyd system in the critical Besov spaces under the ideal case. Furthermore, as a byproduct, we obtain a Beale-Kato-Majida-type criterion (Beale et al., 1984).