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Math. 63:241--258, 2002) to describe vicinal surfaces in heteroepitaxial growth is $$ h_t=-\\left[ H(h_x)+\\left(h_x^{-1}+h_x \\right) h_{xx}\\right]_{xx}, $$ where $h$ denotes the surface height of the film, and $H$ is the Hilbert transform.\n  Existence of solutions was obtained by Dal Maso, Fonseca and Leoni (Arch. Rational Mech. Anal. 212: 1037--1064, 2014). The regularity in time was left unresolved. 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