  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.

Analysis of PDEs