Global Well-posedness for the Three-Dimensional Generalized Phan–Thien–Tanner Model in Critical Besov Spaces

Vanishing limit for the three-dimensional incompressible Phan-Thien–Tanner system

“…In this paper, one of the questions to consider is the vanishing limit for system (PTT) when a tends to zero, so here we assume that 0 a 1. The results of the case a = 0 were proven in some earlier works, see [7,8,10,54] for more details.…”

“…We should also point out that 0 b Ca is a reasonable technical assumption for our limiting problem. In the case a = 0 and b > 0, we see that the solution of system (PTT) will blow up in finite time if the initial data tr τ 0 (x) < 0, which have been discussed in [7,8,10]. Indeed, the authors in [8] only have to consider the initial data tr τ 0 (x) > 0 in the periodic domain T 3 .…”

“…If we assume that τ 0 ∈ H s (R 3 ), then lim x→+∞ τ 0 (x) = 0 which contracts to tr τ 0 (x) > 0. Moreover, under the assumption that tr τ 0 (x) > 0, the authors in [7,10] obtained the global existence of strong solutions provided that the initial data are close to a nontrivial particular solution (depending only on t) in the whole space R 3 . It makes no sense to consider the global well-posedness for system (PTT) with small initial data…”

“…This is another difficult task. To achieve this, we may use a method as in [7,8,10]. We shall make full use of the structure of the constitutive equation of system (PTT) to obtain the time decay estimate of the unknown good quantity tr τ .…”

“…In a more general context, the global well-posedness of the Cauchy problem for the linear and generalized PTT systems in the whole space was achieved by Chen et al. [6, 9] in the critical Besov spaces and [7, 10] in Sobolev spaces. In this paper, we would like to study in a sequel the limit process of the PTT system with when coefficients and go to simultaneously (or, only goes to ) and aim to prove in particular that the obtained limit is the solution of the corresponding Oldroyd-B system.…”