Continuous dependence on the coefficients for a class of non-autonomous evolutionary equations

TL;DR

This paper establishes criteria for the continuous dependence of solutions to non-autonomous evolutionary equations on their coefficients, demonstrating how weak convergence of coefficients leads to weak convergence of solutions, with applications in elasticity, acoustics, and perturbation problems.

Contribution

It provides new criteria linking coefficient convergence to solution convergence for a class of non-autonomous evolutionary equations, including counterexamples for optimality.

Findings

Weak operator topology convergence of coefficients implies weak solution convergence.

Applications demonstrated in elasticity, acoustics, and perturbation problems.

Counterexamples show the optimality of the established criteria.

Abstract

The continuous dependence of solutions to certain (non-autonomous, partial, integro-differential-algebraic, evolutionary) equations on the coefficients is addressed. We give criteria that guarantee that convergence of the coefficients in the weak operator topology implies weak convergence of the respective solutions. We treat three examples: A homogenization problem for a Kelvin-Voigt model for elasticity, the discussion of continuous dependence of the coefficients for acoustic waves with impedance type boundary conditions and a singular perturbation problem for a mixed type equation. By means of counter examples we show optimality of the results obtained.