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

Yann Brenier (1) , Xianglong Duan (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

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.

Mots clés: relative entropy methods; parabolic systems; hyperbolic conservation laws; curve-shortening flow

Domaines: Mathématiques générales [math.GM]

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