Optimal decay rates for solutions to the incompressible Oldroyd-B model in R ³

Consider the Cauchy problem for the incompressible Oldroyd-B model in R ³ . For the case a = 0, global existence results for weak solutions were derived by Lions and Masmoudi (2000 Chin. Ann. Math. B 21 13146), allowing the initial data to be arbitrarily large, whereas it is not known whether this assertion is also true for a -= 0. In this article, time decay estimates for weak solutions subject to arbitrary large data are given for the case a = 0. Furthermore, timedecay estimates are also given for strong solutions for a = 0, however, for small initial data. The decay estimates obtained are of the form that the kth order derivatives in L ² decay as (1 + t)?34 ?k2 for k = 0, 1, 2 as t → ∞. Note that the coupling constant ω does not need to be small throughout this paper.