  Let $X$ be a compact K\"ahler manifold, $E\to X$ a Hermitian vector bundle
and $L\to X$ an ample line bundle. We construct a non-linear heat flow
corresponding to the almost Hermitian-Einstein equation introduced by N.C.
Leung, and prove that the solution exists for a short time. We also construct a
potential function $D_k$ for this flow. In particular, $D_k$ decreases along
the flow.
