Global solutions to the three-dimensional inhomogeneous incompressible Phan-Thien–Tanner system with a class of large initial data

In this paper, we consider the global well-posedness of the three-dimensional inhomogeneous incompressible Phan-Thien–Tanner (PTT) system with general initial data in the critical Besov spaces. The question of whether or not PTT system is globally well posed for large data is still open. When the density ρ is away from zero, we denote by ϱ:=1ρ−1. More precisely, we prove that the PTT system admits a unique global solution, provided that the initial data ϱ ₀, the initial horizontal velocity uh0 , the product ωu30 of the coupling parameter ω and the initial vertical velocity u ₃₀, and the initial symmetric tensor of constraints τ ₀ are sufficiently small. In particular, this result includes the global well-posedness of the PTT system for small initial data in the case where ω∈[0,1) and for large initial vertical velocity in the case where ω tends to zero. As a by-product, our results can be applied to the so-called Oldroyd-B system.