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In this paper, we study the sharp interface limit for solutions of theCahn-Hilliard equation with disparate mobilities. This means that the mobilityfunction degenerates in one of the two energetically favorable configurations,suppressing the diffusion in that phase. First, we construct suitable weaksolutions to this Cahn-Hilliard equation. Second, we prove precompactness ofthese solutions under natural assumptions on the initial data. Third, under anadditional energy convergence assumption, we show that the sharp interfacelimit is a distributional solution to the Hele-Shaw flow with optimalenergy-dissipation rate.

The Hele-Shaw flow as the sharp interface limit of the Cahn-Hilliard equation with disparate mobilities