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

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.

[1]  J. Lions,et al.  Équations Différentielles Opérationnelles Et Problèmes Aux Limites , 1961 .

[2]  R. E. Edwards,et al.  Functional Analysis: Theory and Applications , 1965 .

[3]  Paul L. Butzer,et al.  Semi-groups of operators and approximation , 1967 .

[4]  John Reid,et al.  Semi-Groups of Operators and Approximation , 1969 .

[5]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 2017 .

[6]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[7]  J. Lasry,et al.  Int'egrandes normales et mesures param'etr'ees en calcul des variations , 1973 .

[8]  H. Beckert,et al.  J. L. Lions and E. Magenes, Non‐Homogeneous Boundary Value Problems and Applications, II. (Die Grundlehren d. Math. Wissenschaften, Bd. 182). XI + 242 S. Berlin/Heidelberg/New York 1972. Springer‐Verlag. Preis geb. DM 58,— , 1973 .

[9]  P. Meyer,et al.  Probabilities and potential C , 1978 .

[10]  N. Dinculeanu,et al.  Conditional expectations and weak and strong compactness in spaces of Bochner integrable functions , 1979 .

[11]  Erik J. Balder,et al.  A General Approach to Lower Semicontinuity and Lower Closure in Optimal Control Theory , 1984 .

[12]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[13]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[14]  Stephan Luckhaus,et al.  Solutions for the two-phase Stefan problem with the Gibbs–Thomson Law for the melting temperature , 1990, European Journal of Applied Mathematics.

[15]  Stephan Luckhaus,et al.  Solutions for the Two-Phase Stefan Problem with the Gibbs—Thomson Law for the Melting Temperature , 1990 .

[16]  C. Castaing,et al.  Kolmogorov and Riesz type criteria of compactness in Köthe spaces of vector valued functions , 1990 .

[17]  P. Plotnikov,et al.  Stefan Problem with Surface Tension as a Limit of the Phase Field Model , 1992 .

[18]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[19]  Roger Temam,et al.  Navier–Stokes Equations and Nonlinear Functional Analysis: Second Edition , 1995 .

[20]  Convergence in measure. Local formulation of the Fréchet criterion , 1995 .

[21]  A. Visintin.  Models of Phase Transitions , 1996 .

[22]  Roger Temam,et al.  An optimal compactness theorem and application to elliptic-parabolic systems , 2001, Appl. Math. Lett..

[23]  Compactness properties for families of quasistationary solutions of some evolution equations , 2002 .

[24]  Compactness results for evolution equations , 2004 .

[25]  Jacques Simeon,et al.  Compact Sets in the Space L~(O, , 2005 .