LAUSR

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 $${\mathbb{R}^d}$$Rd, which is a particular case of mean-curvature flow with a 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 and compact, if not empty. Smooth solutions to the curve-shortening flow are always unique in this setting.