Steady state and long time convergence of spirals moving by forced mean curvature motion

Nicolas Forcadel (1) , Cyril Imbert (2) , Régis Monneau (3)

1 LMI - Laboratoire de Mathématiques de l'INSA de Rouen Normandie
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

In this paper, we prove the existence and uniqueness of a ''steady'' spiral moving with forced mean curvature motion. This spiral has a stationary shape and rotates with constant angular velocity. Under appropriate conditions on the initial data, we also show the long time convergence (up to some subsequence in time) of the solution of the Cauchy problem to the steady state. This result is based on a new Liouville result which is of independent interest.

Mots clés: spirals; steady state; mean curvature motion; Liouville theorem; long time convergence; motion of interfaces; viscosity solutions

Domaines: Equations aux dérivées partielles [math.AP]

Nicolas Forcadel, Cyril Imbert, Régis Monneau. Steady state and long time convergence of spirals moving by forced mean curvature motion. 2014. ⟨hal-00975120v1⟩