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Compact sets in the space \(L^ p(0,T;B)\). (English) Zbl 0629.46031

If B denotes a Banach space and if \(T>0\), then C(0,T;B) denotes the Banach space of all continuous functions from [0,T] into B equipped with the uniform convergence norm, and for all \(1\leq p\leq \infty\), \(L^ p(0,T;B)\) is the completion of C(0,T;B) under the norm \(\| f\|_{L^ p(0,T;B)}:=(\int^{T}_{0}\| f(t)\|^ p_ Bdt)^{1/p}\) \((1\leq p<\infty)\) and \(=\| \| f(\cdot)\|_ B\|_{\infty}\) for \(p=\infty.\)
In this paper, the author reviews a number of criteria for a subset F of \(L^ p(0,T;B)\) to be compact.
The criteria presented are in a sense all reformulations of the classical theorem of Gelfand-Phillips-Nakamura stating that a bounded subset of a Banach space is relatively compact if and only if every uniformly bounded net of compact operators that converges pointwise to the identity operator converges uniformly on that subset. In addition, the paper contains a number of sufficient conditions of which the following is typical: If the sequence \(\{f_ n\); \(n=1,2,...\}\) is bounded in \(L^ q(0,T;B)\) and in \(L^ 1_{loc}(0,T;X)\), where \(X\subset B\) and if \(\{\partial f_ n/\partial t\); \(n=1,2,...\}\) is bounded in \(L^ 1_{loc}(0,T;Y)\), where \(B\subset Y\), then for all \(1\leq p<q\), \(\{f_ n\}\) is relatively compact in \(L^ p(0,T;B)\).
Reviewer: W.A.J.Luxemburg

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46A50 Compactness in topological linear spaces; angelic spaces, etc.
46E40 Spaces of vector- and operator-valued functions
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[1] R. A.Adams,Sobolev spaces, Academic Press (1975). · Zbl 0314.46030
[2] Aubin, J. P., Un théorème de compacité, C. R. Acad. Sci., 256, 5042-5044 (1963) · Zbl 0195.13002
[3] J.Bergh - J.Löfstrom,Interpolation Spaces, Springer Verlag (1976), p. 223. · Zbl 0344.46071
[4] Bourbaki, N., Fonctions d’une variable réelle, Act. Sci. Ind. (1958), Paris: Hermann, Paris
[5] Bourbaki, N., Intégration, Act. Sci. Ind. (1965), Paris: Hermann, Paris · Zbl 0136.03404
[6] J. A.Dubinskii,Convergence faible dans les équations elliptiques paraboliques non linéaires, Mat. Sbornik,67, no. 109 (1965).
[7] Grisvard, P., Commutativité de deux foncteurs d’interpolation et applications, Journal de Math., 45, 19-290 (1966) · Zbl 0173.15803
[8] J. L.Lions,Equations différentielles opérationnelles et problèmes aux limites, Springer (1961), p. 111. · Zbl 0098.31101
[9] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires (1969), Paris: Dunod, Paris · Zbl 0189.40603
[10] Lions, J. L.; Magenes, E., Problèmes aux limites non homogènes et applications, vol. 1 et 2 (1968), Paris: Dunod, Paris · Zbl 0165.10801
[11] Lions, J. L.; Magenes, E., Problemi ai limiti non omogenei, III, Annali Scuola Norm. Sup. Pisa, 15, 41-103 (1961)
[12] Lions, J. L.; Peetre, J., Sur une classe d’espace d’interpolation, Inst. Hautes Etudes., 19, 5-68 (1964) · Zbl 0148.11403
[13] J.Necas,Les méthodes directes en théorie des équations elliptiques, Masson (1967). · Zbl 1225.35003
[14] Peetre, J., Espaces d’interpolation et théorème de Sobolev, Ann. Inst. Fourier, 16, 279-317 (1966) · Zbl 0151.17903
[15] Schwartz, L., Théorie des distributions (1951), Parsi: Hermann, Parsi · Zbl 0042.11405
[16] Schwartz, L., Distributions à valeur vectorielles, I, Annales Inst. Fourier, 7, 1-141 (1957)
[17] Simon, J., Ecoulement d’un fluide non homogène avec une densité initiale s’annulant, C. R. Acad. Sci. Paris, 287, 1009-1012 (1978) · Zbl 0395.76038
[18] J.Simon,Remarques sur l’évoulement de fluides non homogènes, Publication du L.A. 189, Université Paris VI (1978).
[19] J.Simon,Fractional Sobolev theorem in one dimension, to appear, preprint of L.A. 189, Univ. Paris VI (1985).
[20] R.Temam,Navier-Stokes equations, North-Holland (1979). · Zbl 0426.35003
[21] E.Temam,Navier-Stokes equations and nonlinear functional analysis, CMBS-NSF, Regional conference series in applied mathematics.
[22] K.Yosida,Functional Analysis, Springer (1965), p. 123. · Zbl 0126.11504
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