We prove the global-in-time existence of weak solution for a hypersurface evolution problem where the velocity is the sum of the mean curvature and arbitrarily given non-smooth vector ﬁeld in a suitable Sobolev space. The approximate solution is obtained by the Allen–Cahn equation with transport term. By establishing the density ratio upper bound on the phase boundary measure it is shown that the limiting surface moves with the desired velocity in the sense of Brakke.
Cite this article
Chun Liu, Norifumi Sato, Yoshihiro Tonegawa, On the existence of mean curvature ﬂow with transport term. Interfaces Free Bound. 12 (2010), no. 2, pp. 251–277DOI 10.4171/IFB/234