Article

Acta Applicandae Mathematicae

, Volume 133, Issue 1, pp 33-43

A Note on Aubin-Lions-Dubinskiĭ Lemmas

  • Xiuqing ChenAffiliated withSchool of Sciences, Beijing University of Posts and TelecommunicationsInstitute for Analysis and Scientific Computing, Vienna University of Technology Email author 
  • , Ansgar JüngelAffiliated withInstitute for Analysis and Scientific Computing, Vienna University of Technology
  • , Jian-Guo LiuAffiliated withDepartment of Physics and Department of Mathematics, Duke University

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Abstract

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.

Keywords

Compactness in Banach spaces Rothe method Dubinskii lemma Seminormed cone

Mathematics Subject Classification (2000)

46B50 35A35