Long-time limit for a class of quadratic infinite-dimensional dynamical systems inspired by models of viscoelastic fluids

TL;DR

This paper investigates quadratic infinite-dimensional dynamical systems inspired by viscoelastic fluid models, proving semi-flow existence, long-term convergence to equilibrium, and discussing potential generalizations.

Contribution

It establishes the semi-flow property and long-term convergence for a class of quadratic systems, extending understanding of their asymptotic behavior.

Findings

Solutions tend to an equilibrium manifold in the $L^2$-norm.

The semi-flow is well-defined on the cone of positive, essentially bounded functions.

Convergence to a specific equilibrium requires additional assumptions.

Abstract

We study a class of quadratic, infinite-dimensional dynamical systems, inspired by models for viscoelastic fluids. We prove that these equations define a semi-flow on the cone of positive, essentially bounded functions. As time tends to infinity, the solutions tend to an equilibrium manifold in the $L^2$-norm. Convergence to a particular function on the equilibrium manifold is only proved under additional assumptions. We discuss several possible generalizations.