Global Well-Posedness and Optimal Time Decay Rates for the Compressible Oldroyd-B Model in ℝ^2

This paper is devoted to the global well-posedness and the optimal time decay rates for the two-dimensional compressible Oldroyd-B model. By introducing new quantities, we find another better perturbation system to eliminate the cross linear terms easily and build the global well-posedness of the strong solution with small initial data in H^2(ℝ^2) -framework. In addition, under the only assumption that the low frequency of the initial data is bounded in Ḃ^-σ_2,∞(ℝ^2) with 0<σ≤ 1 , we obtain the optimal time decay rates of the solution itself and all its derivatives (including the highest spatial derivatives) by using the auxiliary logarithmic decay estimates and analyzing the more accurate decay estimates for the low-high frequency of the solution. Moreover, we treat the cases η̅=0 and η̅>0 in a different way. To prove these results, we shall use the time-weighted energy estimate, Fourier time-splitting and semigroup methods.