Coupled Systems of Differential-Algebraic and Kinetic Equations with Application to the Mathematical Modelling of Muscle Tissue

TL;DR

This paper develops a mathematical framework for coupled differential-algebraic and kinetic equations, proving stability and convergence results, with applications to modeling muscle tissue dynamics involving continuum mechanics and microscopic contraction processes.

Contribution

It introduces a new class of partially kinetic systems combining DAE and kinetic equations, and demonstrates the commutation of index reduction and the mean-field limit.

Findings

Proves Dobrushin's stability estimate for linear systems.

Shows convergence of the mean-field limit to particle dynamics.

Establishes a rigorous link between microscopic and macroscopic models.

Abstract

We consider a coupled system composed of a linear differential-algebraic equation (DAE) and a linear large-scale system of ordinary differential equations where the latter stands for the dynamics of numerous identical particles. Replacing the discrete particles by a kinetic equation for a particle density, we obtain in the mean-field limit the new class of partially kinetic systems. We investigate the influence of constraints on the kinetic theory of those systems and present necessary adjustments. We adapt the mean-field limit to the DAE model and show that index reduction and the mean-field limit commute. As a main result, we prove Dobrushin's stability estimate for linear systems. The estimate implies convergence of the mean-field limit and provides a rigorous link between the particle dynamics and their kinetic description. Our research is inspired by mathematical models for muscle tissue where the macroscopic behaviour is governed by the equations of continuum mechanics, often discretised by the finite element method, and the microscopic muscle contraction process is described by Huxley's sliding filament theory. The latter represents a kinetic equation that characterises the state of the actin-myosin bindings in the muscle filaments. Linear partially kinetic systems are a simplified version of such models, with focus on the constraints.