Weak Operator Continuity for Evolutionary Equations

TL;DR

This paper introduces a new topology for material laws in evolutionary equations that ensures the solution operator is continuous under the weak operator topology, facilitating nonlocal homogenization and modeling in PDE systems.

Contribution

It defines a topology of vector-valued holomorphic functions that lifts the nonlocal H-topology, enabling continuous dependence results for complex PDE models.

Findings

Established a topology ensuring solution operator continuity in evolutionary equations.

Applied the topology to nonlocal homogenization in coupled PDE systems.

Proved continuous dependence for a nonlocal cell migration model.

Abstract

Considering evolutionary equations in the sense of Picard, we identify a certain topology for material laws rendering the solution operator continuous if considered as a mapping from the material laws into the set of bounded linear operators, where the latter are endowed with the weak operator topology. The topology is a topology of vector-valued holomorphic functions and provides a lift of the previously introduced nonlocal $\mathrm{H}$-topology to particular holomorphic functions. The main area of applications are nonlocal homogenisation results for coupled systems of time-dependent partial differential equations. A continuous dependence result for a nonlocal model for cell migration is also provided.