Acta Applicandae Mathematicae

, Volume 133, Issue 1, pp 33–43 | Cite as

A Note on Aubin-Lions-Dubinskiĭ Lemmas

  • Xiuqing Chen
  • Ansgar Jüngel
  • Jian-Guo Liu


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.


Compactness in Banach spaces Rothe method Dubinskii lemma Seminormed cone 

Mathematics Subject Classification (2000)

46B50 35A35 



The first author acknowledges support from the National Science Foundation of China, grant 11101049. The second author acknowledges partial support from the Austrian Science Fund (FWF), grants P20214, P22108, I395, and W1245. This research was supported by the European Union under Grant Agreement number 304617 (Marie-Curie Project “Novel Methods in Computational Finance”). The third author acknowledges support from the National Science Foundation of the USA, grant DMS 10-11738.


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.School of SciencesBeijing University of Posts and TelecommunicationsBeijingChina
  2. 2.Institute for Analysis and Scientific ComputingVienna University of TechnologyWienAustria
  3. 3.Department of Physics and Department of MathematicsDuke UniversityDurhamUSA

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