Mosco convergence framework for singular limits of gradient flows on Hilbert spaces with applications

We consider the question of convergence of a sequence of gradient flows defined on different Hilbert spaces. In order to give meaning to this idea, we introduce a notion of connecting operators. This permits us to generalize the concept of Mosco convergence of functionals to our present setting, and state a desired convergence result for gradient flows, which we then prove. We present a variety of examples, including thin domains, dynamic boundary conditions, and discrete-to-continuum limits.

💡 Research Summary

This paper establishes a comprehensive abstract framework for analyzing the convergence of gradient flows defined on sequences of different Hilbert spaces, a common challenge in singular limit problems such as thin domain reductions, boundary layer asymptotics, and discrete-to-continuum passages.

The core innovation is the introduction of “connecting operators” (L_\varepsilon: X_\varepsilon \to X_0), which are bounded linear maps linking the variable spaces (X_\varepsilon) (where the approximate problems reside) to the fixed limit space (X_0). Using these operators, the authors define what it means for a sequence (w_\varepsilon \in X_\varepsilon) to converge strongly or weakly to an element (w \in X_0) “along (L_\varepsilon)”. They then generalize the classical notion of Mosco convergence for convex functionals to this setting. A sequence of functionals (E_\varepsilon: X_\varepsilon \to