Global existence of weak solutions to some micro-macro models

Abstract

We prove global existence of weak solutions for the co-rotational FENE dumbbell model and the Doi model also called the Rod model. The proof is based on propagation of compactness, namely if we take a sequence of weak solutions which converges weakly and such that the initial data converges strongly then the weak limit is also a solution.To cite this article: A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).Key words Fokker-Planck equations, Navier-Stokes equations, FENE model, Doi model, global solutions. AMS subject classification 35Q30, 82C31, 76A05. RésuméExistence globale de solutions faibles a quelques modèles micro-macro On montre l'existence globale de solutions faiblesà certains modèles micro-macro. En particulier onétudie le modèle FENE (le cas des ressorts) et le modèle de Doi (le cas des barres rigides) La preuve est basée sur la propagation de la compacité.Pour citer cet article : A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).Version française abrégée Le modèle FENEOn montre l'existence globale de solutions faibles au modèle FENE où les polymères sont considérés comme des ressorts avec uneélongation finie. Le systéme est donné par (voir la version anglaise pour les notations).Email addresses: lions@ens.fr (P.-L. Lions), masmoudi@cims.nyu.edu (Nader Masmoudi). Preprint submitted to Elsevier Science 2 mai 2007(1) Le modèle de Doi ou modèle rigideDans le modèle de Doi, les polymères sont représentés par des barres rigides l'orientation R.On démontre le théorème suivant

Well‐posedness for the FENE dumbbell model of polymeric flows

“…Moreover, Lin, Liu and Zhang [24] proved global existence near equilibrium under some restrictions on the potential. Global existence of weak solutions was also proved in [27] for the co-rotational model (see also [2]). …”

On the Euler equations of incompressible fluids

“…PDE models include non-Newtonian viscoelastic models like the Oldroyd-B equations, tensor models, and kinetic models, in which Navier-Stokes equations are coupled to linear or nonlinear Fokker-Planck equations. The well-posedness theory is difficult even in two space dimensions, and consequently the mathematical theory of complex fluids is in its developing stages ( [43], [48], [52], [76], [89], [98], [100], [101], [103], [115]). Some of the models of complex fluids involve stochastic PDEs, or hybrid systems, in which PDEs are coupled to stochastic differential equations.…”

A Note on a Kinetic Model for Rod-Like Particle Suspensions

“…The main purpose of this note is to answer this question by using an assumption of small initial data. See [5][6][7] etc. for more results on Doi related model without considering the effects of gravity.…”