The Motion of closed hypersurfaces in the central force fields

TL;DR

This paper proves large time existence and stability of the evolution of closed hypersurfaces in radially symmetric potentials, modeling charged membranes, using a new Nash-Moser iteration scheme.

Contribution

It introduces a quasi-linear degenerate hyperbolic equation framework and establishes large time existence and stability results for hypersurface motion in general radially symmetric potentials.

Findings

Large time existence of hypersurface evolution established

Stability with respect to small initial data proven

New Nash-Moser iteration scheme developed

Abstract

This paper studies the large time existence for the motion of closed hypersurfaces in a radially symmetric potential. In physical, this surface can be considered as an electrically charged membrane with a constant charge per area in a radially symmetric potential. The evolution of such surface has been investigated by Schn\"urer and Smoczyk (Evolution of hypersurfaces in central force fields, J. Reine Angew. Math. 550 (2002), 77-95). To study its motion, we introduce a quasi-linear degenerate hyperbolic equation which describes the motion of the surfaces extrinsically. Our main results show that the large time existence of such Cauchy problem and the stability with respect to small initial data. When the radially symmetric potential function $v\equiv1$, the local existence and stability results have been obtained by Notz (Closed Hypersurfaces driven by mean curvature and inner pressure, Comm. Pure Appl. Math. 66(5) (2013), 790-819). The proof is based on a new Nash-Moser iteration scheme.