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Weak sequential stability of solutions to a nonisothermal kinetic model for incompressible dilute polymeric fluids

Miroslav Bulíček, Josef Málek, Endre Süli

TL;DR

This work analyzes a thermodynamically consistent nonisothermal kinetic model for incompressible dilute polymeric fluids, coupling the temperature-dependent Navier–Stokes equations with a Fokker–Planck description of polymer configurations and a temperature evolution equation. By deriving the constitutive structure from energy storage and entropy production, the authors obtain explicit expressions for the stress and fluxes, including a polymeric stress $\mathbb{T}$ and a temperature-dependent viscosity $\nu(\theta)$ and conductivity $\kappa(\theta)$. They prove weak sequential stability: any sequence of smooth solutions with uniform bounds converges (up to subsequences) to a global-in-time weak solution that satisfies an energy inequality, with the absolute temperature governed by a renormalized variational inequality; the renormalization is removable if an a priori upper bound on $\theta$ is available. The results lay groundwork toward a rigorous existence theory for this class of nonisothermal kinetic models, highlighting the role of renormalized temperature formulations and the thermodynamic structure in achieving compactness and stability.

Abstract

The paper is concerned with the mathematical analysis of a class of thermodynamically consistent kinetic models for nonisothermal flows of dilute polymeric fluids, based on the identification of energy storage mechanisms and entropy production mechanisms in the fluid under consideration. The model involves a system of nonlinear partial differential equations coupling the unsteady incompressible temperature-dependent Navier--Stokes equations to a temperature-dependent generalization of the classical Fokker--Planck equation and an evolution equation for the absolute temperature. Sequences of smooth solutions to the initial-boundary-value problem, satisfying the available bounds that are uniform with respect to the given data of the model, are shown to converge to a global-in-time large-data weak solution that satisfies an energy inequality, where the absolute temperature satisfies a renormalized variational inequality, implying weak sequential stability of the mathematical model.

Weak sequential stability of solutions to a nonisothermal kinetic model for incompressible dilute polymeric fluids

TL;DR

This work analyzes a thermodynamically consistent nonisothermal kinetic model for incompressible dilute polymeric fluids, coupling the temperature-dependent Navier–Stokes equations with a Fokker–Planck description of polymer configurations and a temperature evolution equation. By deriving the constitutive structure from energy storage and entropy production, the authors obtain explicit expressions for the stress and fluxes, including a polymeric stress and a temperature-dependent viscosity and conductivity . They prove weak sequential stability: any sequence of smooth solutions with uniform bounds converges (up to subsequences) to a global-in-time weak solution that satisfies an energy inequality, with the absolute temperature governed by a renormalized variational inequality; the renormalization is removable if an a priori upper bound on is available. The results lay groundwork toward a rigorous existence theory for this class of nonisothermal kinetic models, highlighting the role of renormalized temperature formulations and the thermodynamic structure in achieving compactness and stability.

Abstract

The paper is concerned with the mathematical analysis of a class of thermodynamically consistent kinetic models for nonisothermal flows of dilute polymeric fluids, based on the identification of energy storage mechanisms and entropy production mechanisms in the fluid under consideration. The model involves a system of nonlinear partial differential equations coupling the unsteady incompressible temperature-dependent Navier--Stokes equations to a temperature-dependent generalization of the classical Fokker--Planck equation and an evolution equation for the absolute temperature. Sequences of smooth solutions to the initial-boundary-value problem, satisfying the available bounds that are uniform with respect to the given data of the model, are shown to converge to a global-in-time large-data weak solution that satisfies an energy inequality, where the absolute temperature satisfies a renormalized variational inequality, implying weak sequential stability of the mathematical model.
Paper Structure (12 sections, 2 theorems, 401 equations)

This paper contains 12 sections, 2 theorems, 401 equations.

Table of Contents

  1. Introduction
  2. Derivation of the model
  3. Formal a priori estimates
  4. Weak sequential stability
  5. Limit passage in the balance of linear momentum equation
  6. Limit passage in the sequence of absolute temperatures
  7. Limit passage in the sequence of probability density functions
  8. Limit passage in the sequence of stress tensors
  9. Limit passage in the Fokker--Planck equation
  10. Limit passage in the renormalized temperature equation
  11. Limit passage in the energy inequality
  12. Conclusions

Key Result

Lemma 1

Let $Q:=(0,T) \times \Omega$ and suppose that $g \in L^1(Q;\mathbb{R}_{> 0})$ and $\varphi \in L^1(Q;W^{1,1}(D;\mathbb{R}_{\geq 0}))$. Suppose further that $U \in C^2([0,\frac{b}{2});\mathbb{R}_{\geq 0})$, $U'$ is nonnegative, $U(s) \to +\infty$ and $U'(s) \to +\infty$ as $s \to (b/2)_{-}$, and that Suppose, finally, that the following bound holds: Then, In particular, $\varphi(t,\boldsymbol{x},

Theorems & Definitions (6)

  • Lemma 1
  • Remark 1
  • Definition 1: Weak solution
  • Theorem 1
  • Remark 2
  • Remark 3