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

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 of mean-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 obtained from the Nambu-Goto action. Using the concept of relative entropy" (or modulated energy"), borrowed from the theory of 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 solutions is convex, compact, if not empty. Smooth solutions to the curve-shortening flow are always unique in this setting.
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Cited literature [19 references]

https://hal.archives-ouvertes.fr/hal-01485459
Contributor : Yann Brenier <>
Submitted on : Wednesday, March 8, 2017 - 7:41:42 PM
Last modification on : Wednesday, March 27, 2019 - 4:10:22 PM
Long-term archiving on : Friday, June 9, 2017 - 2:02:44 PM

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• HAL Id : hal-01485459, version 1
• ARXIV : 1703.03404

### Citation

Yann Brenier, Xianglong Duan. From conservative to dissipative systems through quadratic change of time, with application to the curve-shortening flow. 2017. ⟨hal-01485459⟩

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