  We consider the sharp interface limit of a coupled Stokes/Cahn\textendash
Hilliard system in a two dimensional, bounded and smooth domain, i.e., we
consider the limiting behavior of solutions when a parameter $\epsilon>0$
corresponding to the thickness of the diffuse interface tends to zero. We show
that for sufficiently short times the solutions to the Stokes/Cahn\textendash
Hilliard system converge to solutions of a sharp interface model, where the
evolution of the interface is governed by a Mullins\textendash Sekerka system
with an additional convection term coupled to a two\textendash phase stationary
Stokes system with the Young-Laplace law for the jump of an extra contribution
to the stress tensor, representing capillary stresses. We prove the convergence
result by estimating the difference between the exact and an approximate
solutions. To this end we make use of modifications of spectral estimates shown
by X.\ Chen for the linearized Cahn-Hilliard operator. The treatment of the
coupling terms requires careful estimates, the use of the refinements of the
latter spectral estimate and a suitable structure of the approximate solutions,
which will be constructed in the second part of this contribution.
