From conservative to dissipative systemsthrough 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 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.

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• Hyperbolicity of the time-like extremal surfaces in minkowski spaces
• An integrable example of gradient flows based on optimal transport of differential forms
• Magnetohydrodynamic regime of the born-infeld electromagnetism
• The gradient flow of the potential energy on the space of arcs