Large time behavior to a 2D micro-macro model for compressible polymeric fluids near equilibrium

In this paper, we mainly study the large time behavior to a 2D micro-macro model for compressible polymeric fluids with small initial data. This model is a coupling of isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. Firstly the Fourier splitting method yields that the logarithmic decay rate. By virtue of the time weighted energy estimate, we can improve the decay rate to $(1 + t)^{-\frac{1}{4}}$. Under the low-frequency condition and by the Littlewood-Paley theory, we show that the solutions belong to some Besov spaces with negative index and obtain the optimal $L^2$ decay rate. Finally, we obtain the $\dot{H}^s$ decay rate by establishing a new Fourier splitting estimate

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

In this paper, we mainly study the global strong solutions and its long time decay rates of all order spatial derivatives to a micro-macro model for compressible polymeric fluids 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 micro-macro model admits a unique global strong solution provided the initial data are close to equilibrium state for $d\geq2$. Moreover, for $d\geq3$, we also show a new critical Fourier estimation that allow us to give the long time decay rates of $L^2$ norm for all order spatial derivatives

Global solutions and large time behavior for some Oldroyd-B type models in R2R^2

In this paper, we are concerned with global solutions to the co-rotation Oldroyd-B type model and large time behavior for the general Oldroyd-B type model. We first establish the energy estimate and B-K-M criterion for the 2-D co-rotation Oldroyd-B type model. Then, we obtain global solutions by proving the boundedness of vorticity. In general case, we apply Fourier spiltting method to prove the $H^1$ decay rate for global solutions constructed by T.M.Elgindi and F.Rousset

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

In this paper, we mainly study large time behavior of the strong solutions to the 2-D compressible finite extensible nonlinear elastic (FENE) dumbbell model. The Fourier splitting method yields that the $L^2$ decay rate is $\ln^{-l}(e+t)$ for any $l\in\mathbb{N}$. By virtue of the time weighted energy estimate, we can improve the decay rate to $(1+t)^{-\frac{1}{4}}$. Under the low-frequency condition and by the Littlewood-Paley theory, we show that the solutions belong to some Besov space with negative index and obtain the optimal $L^2$ decay rate.Comment: 22 pages. arXiv admin note: substantial text overlap with arXiv:2107.0864

On diffusive 2D Fokker-Planck-Navier-Stokes systems

We study models kinetic models of polymeric fluids. We introduce a notion of solutions which is based on moments of polymeric distributions. We prove global existence and uniqueness of a large class of initial data for diffusive systems of kinetic equations coupled to fluid equations. As a corollary, we obtain a rigorous derivation of Oldroyd-B closure. We also prove decay of free energy for all the systems considered

Global existence and optimal decay rate of weak solutions to the co-rotation Hooke dumbbell model

In this paper, we mainly study global existence and optimal $L^2$ decay rate of weak solutions to the co-rotation Hooke dumbbell model. This micro-macro model is a coupling of the Navier-Stokes equation with a nonlinear Fokker-Planck equation. Based on the defect measure propagation method, we prove that the co-rotation Hooke dumbbell model admits a global weak solution provided the initial data under different integrability conditions. Moreover, we obtain optimal long time decay rate in $L^2$ for the weak solutions obtained by the Fourier splitting method