  The non-local degenerate Cahn-Hilliard equation is derived from the Vlasov
equation with long-range attraction. We study the local limit as the
delocalization parameter converges to 0. The difficulty arises from the
degeneracy which requires compactness estimates, but all necessary a priori
estimates can be obtained only on the nonlocal quantities yielding almost no
information on the limiting solution itself. We introduce a novel condition on
the nonlocal kernel which allows us to exploit the available nonlocal a priori
estimates. The condition, satisfied by most of the kernels appearing in the
applications, can be of independent interest. Our approach is flexible and
systems can be treated as well.
