# On a degenerate parabolic system describing the mean curvature flow of rotationally symmetric closed surfaces

## TL;DR

This paper reformulates the mean curvature flow of rotationally symmetric closed surfaces as a coupled system of an evolution PDE and ODEs, establishing well-posedness and smoothing effects for the degenerate parabolic problem.

## Contribution

It introduces a new formulation of the mean curvature flow for symmetric surfaces and proves well-posedness and smoothing properties for the resulting nonlinear degenerate parabolic system.

## Key findings

- Well-posedness of the formulated evolution problem
- Parabolic smoothing effect demonstrated
- Coupled PDE-ODE system accurately describes the flow

## Abstract

We show that the mean curvature flow for a closed and rotationally symmetric surface can be formulated as an evolution problem consisting of an evolution equation for the square of the function whose graph is rotated and two ODEs describing the evolution of the points of the evolving surface that lie on the rotation axis. For the fully nonlinear and degenerate parabolic problem we establish the well-posedness property in the setting of classical solutions. Besides we prove that the problem features the effect of parabolic smoothing.