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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Submitted on : Wednesday, March 8, 2017 - 7:41:42 PM
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  • HAL Id : hal-01485459, version 1
  • ARXIV : 1703.03404

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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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