Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 2, p. 395-431

Compactness in the space ${L}^{p}\left(0,T;B\right)$, $B$ 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.

Classification:  28A20,  46E30,  46N20
@article{ASNSP_2003_5_2_2_395_0,
author = {Rossi, Riccarda and Savar\'e, Giuseppe},
title = {Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 2},
number = {2},
year = {2003},
pages = {395-431},
zbl = {1150.46014},
mrnumber = {2005609},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2003_5_2_2_395_0}
}

Rossi, Riccarda; Savaré, Giuseppe. Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 2, pp. 395-431. http://www.numdam.org/item/ASNSP_2003_5_2_2_395_0/

[1] J.P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris 256 (1963), 5042-5044. | MR 152860 | Zbl 0195.13002

[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. | MR 747970 | Zbl 0549.49005

[3] H. Berliocchi - M. Lasry, Intégrandes Normales et Mesures Paramétrées en Calcul de Variations, Bull. Soc. Mat. France 101 (1973), 129-184. | Numdam | MR 344980 | Zbl 0282.49041

[4] H. Brezis, “Analyse fonctionelle - Théorie et applications”, Masson, Paris, 1983. | MR 697382 | Zbl 0511.46001

[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. | MR 548792 | Zbl 0427.46026

[6] P.L. Butzer - H. Berens, "Semi-Groups of Operators and Approximation", Springer, Berlin, 1967. | MR 230022 | Zbl 0164.43702

[7] C. Castaing, Quelques aperçus des résultats de compacité dans ${L}_{E}^{p}$ $\left(1\le p<+\infty \right)$, Travaux Sém. Anal. Convexe 10 (1980), no. 2, exp. no. 16, 25. | MR 620315

[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. | MR 1054796 | Zbl 0714.46025

[9] C. Castaing - M. Valadier, Weak convergence using Young measures, Funct. Approx. Comment. Math. 26 (1998), 7-17. | MR 1666601 | Zbl 0939.28002

[10] C. Dellacherie - P.A. Meyer, “Probabilities and Potential", North-Holland, Amsterdam, 1979. | MR 521810 | Zbl 0494.60001

[11] N. Dunford - J.T. Schwartz, “Linear Operators. Part I", Interscience Publishers, New York, 1958. | MR 117523 | Zbl 0084.10402

[12] R.E. Edwards, “Functional Analysis. Theory and Applications", Holt, Rinehart and Winston, New York, 1965. | MR 221256 | Zbl 0182.16101

[13] L.C. Evans - F. Gariepy, “Measure Theory and Fine Properties of Functions", Studies in Advanced Mathematics, CRC Press, Boca Raton, Florida, 1992. | MR 1158660 | Zbl 0804.28001

[14] J.L. Lions, “Equations différentielles opérationelles et problèmes aux limites", Springer, Berlin, 1961. | Zbl 0098.31101

[15] J.L. Lions, “Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires", Dunod, Gauthiers-Villars, Paris, 1969. | MR 259693 | Zbl 0189.40603

[16] J.L. Lions - E. Magenes, “Non Homogeneous Boundary Value problems and Applications”, volume I, Springer, New York-Heidelberg, 1972. | MR 350177 | Zbl 0223.35039

[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. | MR 1117346 | Zbl 0734.35159

[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. | MR 1236334 | Zbl 0802.35165

[19] J.M. Rakotoson - R. Temam, An Optimal Compactness Theorem and Application to Elliptic-Parabolic Systems, Appl. Math. Lett. 14 (2001), 303-306. | MR 1820617 | Zbl 1001.46049

[20] R. Rossi, Compactness results for evolution equations, Istit. Lombardo Accad. Sci. Lett. Rend. A. 135 (2002), 1-11. | MR 1981626

[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. | MR 1320115 | Zbl 0847.28002

[22] M. Saadoune - M. Valadier, Convergence in measure$:$ the Fréchet criterion from local to global, Bull. Polish Acad. Sci. Math. 43 (1995), 47-57. | MR 1414990 | Zbl 0837.28005

[23] G. Savaré, Compactness Properties for Families of Quasistationary Solutions of some Evolution Equations, to appear in Trans. of A.M.S. | MR 1911517 | Zbl 1008.47065

[24] J. Simon, Compact Sets in the space ${L}^{p}\left(0,T;B\right)$, Ann. Mat. Pura Appl. 146 (1987), 65-96. | MR 916688 | Zbl 0629.46031

[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. | MR 1318914 | Zbl 0833.35110

[26] M. Valdier, Young Measures in Methods of Nonconvex Analysis, Ed. A. Cellina, Lecture Notes in Math. 1446 (Springer-Verlag, Berlin) (1990), 152-188. | MR 1079763 | Zbl 0738.28004

[27] A. Visintin, “Models of Phase Transitions", Birkhäuser, Boston, 1996. | MR 1423808 | Zbl 0882.35004