[Paper Review] Global existence of weak solutions to the FENE dumbbell model of polymeric flows

Nader Masmoudi|arXiv (Cornell University)|Apr 22, 2010

Advanced Mathematical Modeling in Engineering28 references4 citations

TL;DR

This paper establishes the global existence of weak solutions to the FENE dumbbell model of polymeric flows, a system coupling the Navier-Stokes equations with a nonlinear Fokker-Planck equation through micro-macro interactions. The proof relies on weak convergence techniques and a novel transport equation analysis for the defect measure to handle the lack of compactness in the nonlinear stress term, extending existence results to a general class of FENE potentials.

ABSTRACT

Systems coupling fluids and polymers are of great interest in many branches of sciences. One of the models to describe them is the FENE (Finite Extensible Nonlinear Elastic) dumbbell model. We prove global existence of weak solutions to the FENE dumbbell model of polymeric flows for a very general class of potentials. The main problem is the passage to the limit in a nonlinear term that has no obvious compactness properties. The proof uses many weak convergence techniques. In particular it is based on the control of the propagation of strong convergence of some well chosen quantity by studying a transport equation for its defect measure.

Motivation & Objective

• To establish the global existence of weak solutions for the FENE dumbbell model of polymeric flows, which couples fluid dynamics with polymer kinetics.
• To overcome the challenge of lack of compactness in the nonlinear stress term arising from the finite extensibility of polymers near the boundary |R| = R₀.
• To extend the existence theory beyond local solutions by controlling the propagation of strong convergence via a transport equation for the defect measure.
• To provide a rigorous mathematical framework for the micro-macro coupling in polymeric fluids, ensuring energy dissipation and uniform bounds.
• To lay the foundation for future analysis of the zero-diffusion limit and regularity in 2D.

Proposed method

• Use of a regularized approximate system with artificial diffusion in space to ensure existence of smooth solutions.
• Application of weak convergence techniques to pass to the limit in the regularized system as the regularization parameter vanishes.
• Introduction of a defect measure to track the loss of compactness in the nonlinear stress term.
• Derivation and analysis of a transport equation for the defect measure to control its propagation and ensure convergence.
• Use of energy estimates and free-energy dissipation to obtain uniform bounds independent of regularization.
• Employment of Hardy-type inequalities and $L^p$ estimates to control the stress tensor in terms of $H^1$ norms in the polymer configuration variable $R$.

Experimental results

Research questions

• RQ1 Can global weak solutions be constructed for the FENE dumbbell model under general potential assumptions, despite the lack of compactness in the nonlinear stress term?
• RQ2 How can the propagation of strong convergence be controlled in the presence of a defect measure arising from weak convergence?
• RQ3 Is it possible to extend the existence result to the limit of vanishing diffusion in the spatial variable?
• RQ4 Can the global existence result be generalized to other models such as the Hookean or FENE-P models?
• RQ5 Does the micro-macro coupling structure in the FENE model lead to better regularity or stability properties than the Navier-Stokes system?

Key findings

• Global existence of weak solutions is established for the FENE dumbbell model with a general class of FENE-type potentials, including the standard $\mathcal{U}(R) = (1 - |R|^2)^{1-\sigma}$ form.
• The main technical contribution is the control of the defect measure via a transport equation, enabling convergence despite the lack of compactness in the nonlinear term.
• Uniform bounds on the energy and free-energy dissipation are preserved in the limit, ensuring the weak solution satisfies the physical energy balance.
• The method applies to systems with micro-macro interactions and allows for initial data with minimal regularity in the polymer variable $R$.
• The result extends previous local well-posedness results and provides a foundation for studying long-time behavior and regularity.
• The approach is adaptable to other models, such as the FENE-P model, as demonstrated in a follow-up work cited in the paper.