×

Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces. (English) Zbl 1150.46014

In the paper the authors prove a relative compactness criterion in the space \(L^{p}\left( 0,T;B\right)\) and a relative compactness criterion with respect to convergence in measure for \(B\)–valued measurable functions, where \(B\) is a separable Banach space and \(1\leq p<\infty \). The proofs use the Young measure theory. The details are too technical to be stated here.

MSC:

46E40 Spaces of vector- and operator-valued functions
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
46N20 Applications of functional analysis to differential and integral equations
PDFBibTeX XMLCite
Full Text: EuDML

References:

[1] J.P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris 256 (1963), 5042-5044. Zbl0195.13002 MR152860 · 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. Zbl0549.49005 MR747970 · 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. Zbl0282.49041 MR344980 · Zbl 0282.49041
[4] H. Brezis, “Analyse fonctionelle - Théorie et applications”, Masson, Paris, 1983. Zbl0511.46001 MR697382 · 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. Zbl0427.46026 MR548792 · Zbl 0427.46026
[6] P.L. Butzer - H. Berens, ”Semi-Groups of Operators and Approximation”, Springer, Berlin, 1967. Zbl0164.43702 MR230022 · Zbl 0164.43702
[7] C. Castaing, Quelques aperçus des résultats de compacité dans \(L^{p}_{E}\) \((1\leq p&lt;+\infty )\), Travaux Sém. Anal. Convexe 10 (1980), no. 2, exp. no. 16, 25. MR620315
[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. Zbl0714.46025 MR1054796 · Zbl 0714.46025
[9] C. Castaing - M. Valadier, Weak convergence using Young measures, Funct. Approx. Comment. Math. 26 (1998), 7-17. Zbl0939.28002 MR1666601 · Zbl 0939.28002
[10] C. Dellacherie - P.A. Meyer, “Probabilities and Potential”, North-Holland, Amsterdam, 1979. Zbl0494.60001 MR521810 · Zbl 0494.60001
[11] N. Dunford - J.T. Schwartz, “Linear Operators. Part I”, Interscience Publishers, New York, 1958. Zbl0084.10402 MR117523 · Zbl 0084.10402
[12] R.E. Edwards, “Functional Analysis. Theory and Applications”, Holt, Rinehart and Winston, New York, 1965. Zbl0182.16101 MR221256 · 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. Zbl0804.28001 MR1158660 · Zbl 0804.28001
[14] J.L. Lions, “Equations différentielles opérationelles et problèmes aux limites”, Springer, Berlin, 1961. Zbl0098.31101 · Zbl 0098.31101
[15] J.L. Lions, “Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires”, Dunod, Gauthiers-Villars, Paris, 1969. Zbl0189.40603 MR259693 · Zbl 0189.40603
[16] J.L. Lions - E. Magenes, “Non Homogeneous Boundary Value problems and Applications”, volume I, Springer, New York-Heidelberg, 1972. Zbl0223.35039 MR350177 · 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. Zbl0734.35159 MR1117346 · 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. Zbl0802.35165 MR1236334 · 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. Zbl1001.46049 MR1820617 · Zbl 1001.46049
[20] R. Rossi, Compactness results for evolution equations, Istit. Lombardo Accad. Sci. Lett. Rend. A. 135 (2002), 1-11. MR1981626 · Zbl 1167.35556
[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. Zbl0847.28002 MR1320115 · 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. Zbl0837.28005 MR1414990 · 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. Zbl1008.47065 MR1911517 · Zbl 1008.47065
[24] J. Simon, Compact Sets in the space \( L^{p}(0,T;B) \), Ann. Mat. Pura Appl. 146 (1987), 65-96. Zbl0629.46031 MR916688 · 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. Zbl0833.35110 MR1318914 · 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. Zbl0738.28004 MR1079763 · Zbl 0738.28004
[27] A. Visintin, “Models of Phase Transitions”, Birkhäuser, Boston, 1996. Zbl0882.35004 MR1423808 · Zbl 0882.35004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.