Singular perturbation results for linear partial differential-algebraic equations of hyperbolic type

TL;DR

This paper investigates the behavior of hyperbolic partial differential-algebraic equations with a small parameter, analyzing how solutions transition to parabolic equations as the parameter approaches zero, using variational methods and estimates.

Contribution

It provides a rigorous analysis of singular perturbation limits for hyperbolic PDE-DAEs, including well-posedness and solution estimates for various initial data.

Findings

First- and second-order estimates for specific initial data

Lower-order estimates for general initial data

Numerical validation of estimate optimality

Abstract

We consider constrained partial differential equations of hyperbolic type with a small parameter $\varepsilon>0$, which turn parabolic in the limit case, i.e., for $\varepsilon=0$. The well-posedness of the resulting systems is discussed and the corresponding solutions are compared in terms of the parameter $\varepsilon$. For the analysis, we consider the system equations as partial differential-algebraic equation based on the variational formulation of the problem. For a particular choice of the initial data, we reach first- and second-order estimates. For general initial data, lower-order estimates are proven and their optimality is shown numerically.