# From conservative to dissipative systemsthrough quadratic change of time, with application to the curve-shortening flow

## TL;DR

This paper demonstrates how dissipative systems like the curve-shortening flow can be derived from conservative systems via a quadratic change of time, introducing a new framework of generalized solutions using relative entropy.

## Contribution

It introduces a method to obtain dissipative flows from conservative models through quadratic time change and defines dissipative solutions for the curve-shortening flow.

## Key findings

- Dissipative systems can be derived from conservative ones via quadratic time change.
- A new notion of dissipative solutions for the curve-shortening flow is proposed.
- Smooth solutions are unique within the set of generalized solutions.

## Abstract

We provide several examples of dissipative systems that can be obtained from conservative ones through a simple, quadratic,change of time. A typical example is the curve-shortening flow in R^d, which is a particular case ofmean-curvature flow with co-dimension higher than one (except in the case d=2).Through such a change of time, this flow can be formally derived from the conservative model of vibrating strings obtainedfrom the Nambu-Goto action. Using the concept of "relative entropy" (or "modulated energy"), borrowed from the theoryof hyperbolic systems of conservation laws, we introduce a notion of generalized solutions,that we call dissipative solutions, for the curve-shortening flow. For given initial conditions, the set of generalized solutionsis convex, compact, if not empty. Smooth solutions to the curve-shortening flow are always unique in this setting.