Global strong solutions and large time behavior to the compressible co-rotation FENE dumbbell model of polymeric flows near equilibrium

Abstract

In this paper, we mainly study the global well-posedness and L 2 decay rate for the strong solutions of the compressible co-rotation finite extensible nonlinear elastic (FENE) dumbbell model with small initial data. This model is a coupling of isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. We first prove that the FENE dumbbell model admits a unique global strong solution provided the initial data are close to equilibrium state for d ≥ 2. Moreover, for d ≥ 3, we show that optimal L 2 decay rates of global strong solutions by the linear spectral theory.

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

“…Proof. By virtue of the coupling effect between (ρ, u, g) and using the energy methods, we can easily deduce that the lower order derivatives estimates for (1.2), see [27].…”

“…We divide the proof of Theorem 1.1 into two Propositions. Using the standard iterating method in [18] and [27], one can easily deduce that the existence of local solutions. Thus we omit the proof here and present the following Proposition.…”

“…Recently, N. Masmoudi [30] is concerning with the long time behavior for polymeric models. The co-rotation compressible FENE system has been studied in [27]. In the co-rotation case, the drag term σ(u) = ∇u−∇u T 2 which leads to some good structure such that the ψ − ψ ∞ is exponential decay in time.…”

“…Using the energy methods in [27], one can deduce that the global existence of strong solutions for (1.2). However, to obtain optimal time decay rate, we need a more precise higher order derivatives estimate for (1.2).…”

“…Moreover the potential U(R) = −k log(1−( |R| |R0| ) 2 ) for some constant k > 0. σ(u) = ∇u is the drag term. This is a micro-macro model (For more details, one can refer to [27], [28] and [29]).…”