The Optimal Time Decay Rates and Vanishing Limit for Three-Dimensional Incompressible Phan-Thien–Tanner and Oldroyd-B Systems in the Critical Besov Spaces

In this article, we study the incompressible Phan-Thien–Tanner (PTT) and Oldroyd-B systems in R 3 . The interesting features of the Cauchy problem studied in this work are the well-posedness and large-time behavior of global solutions for the PTT and Oldroyd-B systems with or without a damping mechanism, and the relationship between these two systems in the critical Besov spaces. First, we study the well-posedness of global solutions with data having critical regularity in the case of small initial data by proving uniform estimates with respect to the parameters a and b . Second, we prove the optimal time decay rates of global solutions in L q by exploiting harmonic analysis tools, such as non-standard L q product estimates, various Sobolev embeddings, and interpolation inequalities. We only assume that the negative Besov norm at low frequencies of the initial data is bounded and remove the L p ( 1 ≤ p < 2 ) condition, which has been required in previous works on this problem. Finally, we investigate the convergence of global solutions to the PTT system when b tends to 0 . Moreover, the specific rate of convergence is obtained in some sense.