# Interfaces and Free Boundaries

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**Volume 12, Issue 2, 2010, pp. 251–277**

**DOI: 10.4171/IFB/234**

On the existence of mean curvature ﬂow with transport term

Chun Liu^{[1]}, Norifumi Sato

^{[2]}and Yoshihiro Tonegawa

^{[3]}(1) Department of Mathematics, The Pennsylvania State University, 207 Church Street SE, PA 16802, UNIVERSITY PARK, UNITED STATES

(2) Furano H.S., 076-0011, FURANO (HOKKAIDO), JAPAN

(3) Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-ku, 060-0810, SAPPORO, JAPAN

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.

*Keywords: *Mean curvature ﬂow, varifold, Allen–Cahn equation, phase ﬁeld method

Liu C, Sato N, Tonegawa Y. On the existence of mean curvature ﬂow with transport term. *Interfaces Free Bound.* 12 (2010), 251-277. doi: 10.4171/IFB/234