Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces

Riccarda Rossi; Giuseppe Savaré

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 2, page 395-431
  • ISSN: 0391-173X

Abstract

top
Compactness in the space L p ( 0 , T ; B ) , B being a separable Banach space, has been deeply investigated by J.P. Aubin (1963), J.L. Lions (1961, 1969), J. Simon (1987), and, more recently, by J.M. Rakotoson and R. Temam (2001), who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to several abstract time dependent problems related to evolutionary PDEs. In the present paper, the problem is examined in view of Young measure theory: exploiting the underlying principles of “tightness” and “integral equicontinuity”, new necessary and sufficient conditions for compactness are given, unifying some of the previous contributions and showing that the Aubin - Lions condition is not only sufficient but also necessary for compactness. Furthermore, the related issue of compactness with respect to convergence in measure is studied and a general criterion is proved.

How to cite

top

Rossi, Riccarda, and Savaré, Giuseppe. "Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.2 (2003): 395-431. <http://eudml.org/doc/84506>.

@article{Rossi2003,
abstract = {Compactness in the space $\{L^\{p\}(0,T;B)\}$, $B$ being a separable Banach space, has been deeply investigated by J.P. Aubin (1963), J.L. Lions (1961, 1969), J. Simon (1987), and, more recently, by J.M. Rakotoson and R. Temam (2001), who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to several abstract time dependent problems related to evolutionary PDEs. In the present paper, the problem is examined in view of Young measure theory: exploiting the underlying principles of “tightness” and “integral equicontinuity”, new necessary and sufficient conditions for compactness are given, unifying some of the previous contributions and showing that the Aubin - Lions condition is not only sufficient but also necessary for compactness. Furthermore, the related issue of compactness with respect to convergence in measure is studied and a general criterion is proved.},
author = {Rossi, Riccarda, Savaré, Giuseppe},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {395-431},
publisher = {Scuola normale superiore},
title = {Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces},
url = {http://eudml.org/doc/84506},
volume = {2},
year = {2003},
}

TY - JOUR
AU - Rossi, Riccarda
AU - Savaré, Giuseppe
TI - Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2003
PB - Scuola normale superiore
VL - 2
IS - 2
SP - 395
EP - 431
AB - Compactness in the space ${L^{p}(0,T;B)}$, $B$ being a separable Banach space, has been deeply investigated by J.P. Aubin (1963), J.L. Lions (1961, 1969), J. Simon (1987), and, more recently, by J.M. Rakotoson and R. Temam (2001), who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to several abstract time dependent problems related to evolutionary PDEs. In the present paper, the problem is examined in view of Young measure theory: exploiting the underlying principles of “tightness” and “integral equicontinuity”, new necessary and sufficient conditions for compactness are given, unifying some of the previous contributions and showing that the Aubin - Lions condition is not only sufficient but also necessary for compactness. Furthermore, the related issue of compactness with respect to convergence in measure is studied and a general criterion is proved.
LA - eng
UR - http://eudml.org/doc/84506
ER -

References

top
  1. [1] J.P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris 256 (1963), 5042-5044. Zbl0195.13002MR152860
  2. [2] E.J. Balder, A General Approach to Lower Semicontinuity and Lower Closure in Optimal Control Theory, SIAM J. Control Optim. 22 (1984 a), 570-568. Zbl0549.49005MR747970
  3. [3] H. Berliocchi – M. Lasry, Intégrandes Normales et Mesures Paramétrées en Calcul de Variations, Bull. Soc. Mat. France 101 (1973), 129-184. Zbl0282.49041MR344980
  4. [4] H. Brezis, “Analyse fonctionelle - Théorie et applications”, Masson, Paris, 1983. Zbl0511.46001MR697382
  5. [5] J.K. Brooks – N. Dinculeanu, Conditional expectations and weak and strong compactness in spaces of Bochner integrable functions, J. Multivariate Anal. 9 (1979), 420-427. Zbl0427.46026MR548792
  6. [6] P.L. Butzer – H. Berens, "Semi-Groups of Operators and Approximation", Springer, Berlin, 1967. Zbl0164.43702MR230022
  7. [7] C. Castaing, Quelques aperçus des résultats de compacité dans L E p ( 1 ≤ p l t ; + ∞ ) , Travaux Sém. Anal. Convexe 10 (1980), no. 2, exp. no. 16, 25. MR620315
  8. [8] C. Castaing – A. Kaminska, Kolmogorov and Riesz type criteria of compactness in Köthe spaces of vector valued functions, J. Math. Anal. Appl. 149 (1990), 96–113. Zbl0714.46025MR1054796
  9. [9] C. Castaing – M. Valadier, Weak convergence using Young measures, Funct. Approx. Comment. Math. 26 (1998), 7–17. Zbl0939.28002MR1666601
  10. [10] C. Dellacherie – P.A. Meyer, “Probabilities and Potential", North-Holland, Amsterdam, 1979. Zbl0494.60001MR521810
  11. [11] N. Dunford – J.T. Schwartz, “Linear Operators. Part I", Interscience Publishers, New York, 1958. Zbl0084.10402MR117523
  12. [12] R.E. Edwards, “Functional Analysis. Theory and Applications", Holt, Rinehart and Winston, New York, 1965. Zbl0182.16101MR221256
  13. [13] L.C. Evans – F. Gariepy, “Measure Theory and Fine Properties of Functions", Studies in Advanced Mathematics, CRC Press, Boca Raton, Florida, 1992. Zbl0804.28001MR1158660
  14. [14] J.L. Lions, “Equations différentielles opérationelles et problèmes aux limites", Springer, Berlin, 1961. Zbl0098.31101
  15. [15] J.L. Lions, “Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires", Dunod, Gauthiers-Villars, Paris, 1969. Zbl0189.40603MR259693
  16. [16] J.L. Lions – E. Magenes, “Non Homogeneous Boundary Value problems and Applications”, volume I, Springer, New York-Heidelberg, 1972. Zbl0223.35039MR350177
  17. [17] S. Luckhaus, Solutions of the two phase Stefan problem with the Gibbs-Thomson law for the melting temperature, Euro. J. Appl. Math. 1 (1990), 101-111. Zbl0734.35159MR1117346
  18. [18] P.I. Plotnikov – V.N. Starovoitov, The Stefan problem with surface tension as a limit of phase field model, Differential Equations 29 (1993), 395-404. Zbl0802.35165MR1236334
  19. [19] J.M. Rakotoson – R. Temam, An Optimal Compactness Theorem and Application to Elliptic-Parabolic Systems, Appl. Math. Lett. 14 (2001), 303-306. Zbl1001.46049MR1820617
  20. [20] R. Rossi, Compactness results for evolution equations, Istit. Lombardo Accad. Sci. Lett. Rend. A. 135 (2002), 1-11. Zbl1167.35556MR1981626
  21. [21] M. Saadoune – M. Valadier, Convergence in measure. Local formulation of the Fréchet criterion, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 423–428. Zbl0847.28002MR1320115
  22. [22] M. Saadoune – M. Valadier, Convergence in measure : the Fréchet criterion from local to global, Bull. Polish Acad. Sci. Math. 43 (1995), 47–57. Zbl0837.28005MR1414990
  23. [23] G. Savaré, Compactness Properties for Families of Quasistationary Solutions of some Evolution Equations, to appear in Trans. of A.M.S. Zbl1008.47065MR1911517
  24. [24] J. Simon, Compact Sets in the space L p ( 0 , T ; B ) , Ann. Mat. Pura Appl. 146 (1987), 65-96. Zbl0629.46031MR916688
  25. [25] R. Temam, Navier-Stokes equations and nonlinear functional analysis. Second edition, CBMS-NSF Regional Conference Series in Applied Mathematics 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. Zbl0833.35110MR1318914
  26. [26] M. Valdier, Young Measures in Methods of Nonconvex Analysis, Ed. A. Cellina, Lecture Notes in Math. 1446 (Springer-Verlag, Berlin) (1990), 152-188. Zbl0738.28004MR1079763
  27. [27] A. Visintin, “Models of Phase Transitions", Birkhäuser, Boston, 1996. Zbl0882.35004MR1423808

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.