Global strong solutions and optimal $L^2$ decay to the compressible FENE dumbbell model

Abstract

In this paper, we are concerned with the global well-posedness and L 2 decay rate for the strong solutions of the compressible finite extensible nonlinear elastic (FENE) dumbbell model. For d ≥ 2, we prove that the compressible FENE dumbbell model admits a unique global strong solution provided the initial data are close to equilibrium state. Moreover, by the Littlewood-Paley decomposition theory and the Fourier splitting method, we show optimal L 2 decay rate of global strong solutions for d ≥ 3.

Global strong solutions and large time behavior to a micro-macro model for compressible polymeric fluids near equilibrium

“…We divide it into three steps to prove Proposition 3.2. The estimates in the first two steps is similar to those in [16] with different external force τ . It should be underlined that such difference results in additional estimates for mixed derivatives as shown in the third step.…”

Large time behavior of global strong solutions to the 2-D compressible FENE dumbbell model

“…They proved the global well-posedness for (1.1) if the initial data is close to the equilibrium and R ∈ R 3 . For the compressible FENE system, Z. Luo, W. Luo and Z. Yin [30] proved the global well-posedness results with d ≥ 2 and studied large time behaviour with d ≥ 3.…”

“…Recently, N. Masmoudi [33] is concerning with the long time behavior for polymeric models. Large time behaviour of the compressible FENE system with d ≥ 3 has been studied in [29] and [30]. To our best knowledge, for d = 2, large time behaviour for the compressible FENE system (1.2) has not been studied yet.…”

“…Theorem 2.6. [30] Let d = 2 and s > 2. Let (ρ, u, g) be a strong solution of (1.2) with the initial data (ρ 0 , u 0 , g 0 ) satisfying the conditions B g 0 ψ ∞ dR = 0 and 1 + g 0 > 0.…”