A Note on Aubin-Lions-Dubinskiĭ Lemmas

Strong compactness results for families of functions in seminormed nonnegative cones in the spirit of the Aubin-Lions-Dubinskiĭ lemma are proven, refining some recent results in the literature. The first theorem sharpens slightly a result of Dubinskiĭ (in Mat. Sb. 67(109):609–642, 1965) for seminormed cones. The second theorem applies to piecewise constant functions in time and sharpens slightly the results of Dreher and Jüngel (in Nonlinear Anal. 75:3072–3077, 2012) and Chen and Liu (in Appl. Math. Lett. 25:2252–2257, 2012). An application is given, which is useful in the study of porous-medium or fast-diffusion type equations.