Consider the Cauchy problem for the incompressible Oldroyd-B model in . For the case , global existence results for weak solutions were derived by Lions and Masmoudi (2000 Chin. Ann. Math . B 21 131–46), allowing the initial data to be arbitrarily large, whereas it is not known whether this assertion is also true for . In this article, time decay estimates for weak solutions subject to arbitrary large data are given for the case . Furthermore, time-decay estimates are also given for strong solutions for , however, for small initial data. The decay estimates obtained are of the form that the th order derivatives in decay as for as . Note that the coupling constant does not need to be small throughout this paper.

In this paper, we consider the Cauchy problem for an inviscid compressible Oldroyd-B model in three dimensions. The global well posedness of strong solutions and the associated time-decay estimates in Sobolev spaces are established near an equilibrium state. The vanishing of viscosity is the main challenge compared with [47] where the viscosity coefficients are included and the decay rates for the highest-order derivatives of the solutions seem not optimal. One of the main objectives of this paper is to develop some new dissipative estimates such that the smallness of the initial data and decay rates are independent of the viscosity. Moreover, we prove that the decay rates for the highest-order derivatives of the solutions are optimal, which is of independent interest. Our proof relies on Fourier theory and delicate energy method.