Global regularity and sharp decay for incompressible Oldroyd-B model with only fractional dissipation of stress tensor

In this paper, we investigate the Cauchy problem for the d-dimensional ([Formula: see text]) incompressible Oldroyd-B model with only fractional dissipation [Formula: see text], [Formula: see text]. Applying the pure energy argument rather than spectral analysis method, we obtain the small initial data global existence results in critical Besov spaces, and establish the sharp time-decay rates of the global solutions to such model for the case [Formula: see text] and the case [Formula: see text], respectively. In particular, our results include the situation that Oldroyd-B model possesses only stress tensor damping and has no dissipation term.