# From Visco-Energetic to Energetic and Balanced Viscosity solutions of rate-independent systems

## TL;DR

This paper investigates Visco-Energetic solutions for rate-independent systems, showing they serve as an intermediate between Energetic and Balanced Viscosity solutions, with convergence results as the viscous parameter varies.

## Contribution

It formalizes the intermediate nature of Visco-Energetic solutions and proves their convergence to Energetic or Balanced Viscosity solutions as the viscous parameter approaches zero or infinity.

## Key findings

- Convergence to Energetic solutions as viscous parameter approaches zero.
- Convergence to Balanced Viscosity solutions as viscous parameter approaches infinity.
- Provides a unified framework connecting different weak solution concepts.

## Abstract

This paper focuses on weak solvability concepts for rate-independent systems in a metric setting. Visco-Energetic solutions have been recently obtained by passing to the time-continuous limit in a time-incremental scheme, akin to that for Energetic solutions, but perturbed by a `viscous' correction term, as in the case of Balanced Viscosity solutions. However, for Visco-Energetic solutions this viscous correction is tuned by a fixed parameter $\mu$. The resulting solution notion is characterized by a stability condition and an energy balance analogous to those for Energetic solutions, but, in addition, it provides a fine description of the system behavior at jumps as Balanced Viscosity solutions do. Visco-Energetic evolution can be thus thought as `in-between' Energetic and Balanced Viscosity evolution. Here we aim to formalize this intermediate character of Visco-Energetic solutions by studying their singular limits as $\mu\downarrow 0$ and $\mu\uparrow \infty$. We shall prove convergence to Energetic solutions in the former case, and to Balanced Viscosity solutions in the latter situation.