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

Abstract

In this article, we study the incompressible Phan-Thien–Tanner (PTT) and Oldroyd-B systems in ${\mathbb{R}}^3$ . 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 $L^p$ ( $1\le p<2$ ) 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.