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On the well-posedness of a nonlocal kinetic model for dilute polymers with anomalous diffusion

Marvin Fritz, Endre Süli, Barbara Wohlmuth

TL;DR

This work establishes global-in-time existence and, under regularity, uniqueness of weak solutions to a nonlocal-in-time Navier–Stokes–Fokker–Planck system modeling dilute polymers with memory. The authors employ energy–entropy methods, maximum-principle arguments for nonnegativity, and a novel nonlocal Aubin–Lions compactness framework to handle memory effects driven by a Prabhakar–Caputo type kernel. A multi-level Galerkin truncation scheme is developed, with careful passes to the limits in velocity and configuration-space discretizations, ensuring strong convergence and nonnegativity of the probability density. The results provide a rigorous well-posedness foundation for subdiffusive polymeric flows with memory and suggest avenues for structure-preserving numerical schemes and future research on the impact of different temporal derivatives.

Abstract

In this work, we study a class of nonlocal-in-time kinetic models of incompressible dilute polymeric fluids. The system couples a macroscopic balance of linear momentum equation with a mezoscopic subdiffusive Fokker-Planck equation governing the evolution of the probability density function of polymer configurations. The model incorporates nonlocal features to capture subdiffusive and memory-type phenomena. Our main result asserts the existence of global-in-time large-data weak solutions to this nonlocal system. The proof relies on an energy estimate involving a suitable relative entropy, which enables us to handle the critical general non-corotational drag term that couples the two equations. As a side result, we prove nonnegativity of the probability density function. A crucial step in our analysis is to establish strong convergence of the sequence of Galerkin approximations by a combination of techniques, involving a novel compactness result for nonlocal PDEs. Lastly, we prove the uniqueness of weak solutions with sufficient regularity.

On the well-posedness of a nonlocal kinetic model for dilute polymers with anomalous diffusion

TL;DR

This work establishes global-in-time existence and, under regularity, uniqueness of weak solutions to a nonlocal-in-time Navier–Stokes–Fokker–Planck system modeling dilute polymers with memory. The authors employ energy–entropy methods, maximum-principle arguments for nonnegativity, and a novel nonlocal Aubin–Lions compactness framework to handle memory effects driven by a Prabhakar–Caputo type kernel. A multi-level Galerkin truncation scheme is developed, with careful passes to the limits in velocity and configuration-space discretizations, ensuring strong convergence and nonnegativity of the probability density. The results provide a rigorous well-posedness foundation for subdiffusive polymeric flows with memory and suggest avenues for structure-preserving numerical schemes and future research on the impact of different temporal derivatives.

Abstract

In this work, we study a class of nonlocal-in-time kinetic models of incompressible dilute polymeric fluids. The system couples a macroscopic balance of linear momentum equation with a mezoscopic subdiffusive Fokker-Planck equation governing the evolution of the probability density function of polymer configurations. The model incorporates nonlocal features to capture subdiffusive and memory-type phenomena. Our main result asserts the existence of global-in-time large-data weak solutions to this nonlocal system. The proof relies on an energy estimate involving a suitable relative entropy, which enables us to handle the critical general non-corotational drag term that couples the two equations. As a side result, we prove nonnegativity of the probability density function. A crucial step in our analysis is to establish strong convergence of the sequence of Galerkin approximations by a combination of techniques, involving a novel compactness result for nonlocal PDEs. Lastly, we prove the uniqueness of weak solutions with sufficient regularity.

Paper Structure

This paper contains 14 sections, 14 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Modelling
  5. Function spaces
  6. Nonlocal derivative
  7. A generalisation of the Aubin--Lions lemma with nonlocal derivatives
  8. Analysis
  9. Step 1: Truncation and Galerkin approximation
  10. Step 2: Uniform a priori estimates in $n$ and the limit process $n \to \infty$
  11. Step 3: Uniform a priori estimates in $m$ and the limit process $m \to \infty$
  12. Step 4: Uniform a priori estimates with respect to the truncation parameter $\ell$ and the final limit process $\ell \to \infty$
  13. Step 5: Uniqueness
  14. Outlook

Key Result

Lemma 1

Let $X \hookrightarrow\space\hookrightarrow Y \hookrightarrow Z$ be real Banach spaces, $p \in [1,\infty)$ and $T>0$. If $\mathcal{G}$ is a bounded subset of $L^p(0,T;X) \cap BV(0,T;Z)$, then $\mathcal{G}$ is relatively compact in $L^p(0,T;Y)$.

Theorems & Definitions (29)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • proof
  • Definition 1
  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • Lemma 3
  • ...and 19 more