  We show convergence of solutions of a convective Allen-Cahn equation for a
given smooth and divergence free velocity field to a transport equation for an
evolving interface in the case when the thickness of the diffuse interface
tends to zero and the mobility coefficient is proportional to the interfacial
thickness. This is done for well-prepared initial data by estimating the
difference of the exact and an approximate solution of the convective
Allen-Cahn equation with the aid of a uniform lower bound for the linearized
Allen-Cahn operator. The approximate solution is constructed with the aid of
three terms from formally matched asymptotics calculations close to the
interface.
