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

Abstract

We consider a coupled system composed of a differential-algebraic equation and a large-scale ordinary differential equation where the latter stands for the dynamics of numerous identical particles. Replacing the discrete particles by a kinetic equation in terms of the particle density, we obtain a new class of models that we refer to as partially kinetic systems. We investigate the influence of constraints on the kinetic theory of those systems and present necessary adjustments. An essential tool from kinetic theory, the mean-field limit, also applies to partially kinetic systems as well, which yields a rigorous link between the kinetic equations and their underlying particle dynamics. Our research is inspired by the mathematical models for muscle tissue where the macroscopic behavior is governed by the equations of continuum mechanics, often discretized 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 characterizes the state of the actin-myosin bindings in the muscle filaments. As a prime example, we analyse the influence of constraints on the kinetic theory of a simplified version of Huxley’s sliding filament model. The general theory of partially kinetic systems is in its early stages. We introduce the equations of motions for partially kinetic systems, which is family of differential-algebraic equations. We conjecture that classical proofs from kinetic theory for global existence and Dobrunshin’s stability estimate can be adjusted for partially kinetic systems.