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

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Compactness in the space , 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

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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

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