# Existence of weak solution for mean curvature flow with transport term and forcing term

## TL;DR

This paper proves the global existence of weak solutions for mean curvature flow with non-smooth transport and forcing terms using a modified Allen-Cahn equation, advancing understanding of geometric flows with external influences.

## Contribution

It establishes the existence of weak solutions for mean curvature flow with non-smooth terms, employing a modified Allen-Cahn approach and monotonicity formula techniques.

## Key findings

- Proved global existence of weak solutions.
- Applied modified Allen-Cahn equation with useful properties.
- Extended analysis to non-smooth transport and forcing terms.

## Abstract

We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove the global existence of the weak solutions for the mean curvature flow with the terms, by using the modified Allen-Cahn equation that holds useful properties such as the monotonicity formula.