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

TL;DR

This paper proves the weak sequential stability of solutions for a complex nonisothermal kinetic model describing dilute polymeric fluids, ensuring solutions converge under certain bounds.

Contribution

It establishes the global-in-time weak solution existence and stability for a coupled thermodynamically consistent kinetic model involving Navier-Stokes and Fokker-Planck equations.

Findings

Sequences of smooth solutions converge to a global weak solution.

The weak solution satisfies an energy inequality.

The temperature component satisfies a renormalized variational inequality.

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.