Semiflows Associated with Compact and Uniform Processes by CONSTANTINE M. DAFERMOS* Center for Dynamical Systems Division of Applied Mathematics Brown University Providence, Rhode Island 02912 1. Introduction. LaSalle's invariance principle [7] provides information on the structure of oJ-limit sets in autonomous systems of ordinary differential equations endowed with a Liapunov function, and consequently is a powerful tool for studying the asymptotic behavior of solutions. The principle, extended in many directions, has been applied to a wide variety of evolutionary equations (e.g., Hale [6]). In an effort to collect the known results under a unifying theory and at the same time make possible novel applications, the author [2] has established an invariance principle for compact processes. It was demonstrated (Sell [9]) that processes generated by nonautonomous systems of ordinary differential equations can be visualized as dynamical systems on an appropriate phase space. A similar viewpoint has been adopted by Miller and Sell [8] for processes generated by Volterra integral equations. Motivated by the above work, we show in Section 2 that every process can be visualized in a canonical way as a semiflow. Our program is to deduce properties of oJ-limit sets in a compact process from properties of w-limit sets in the associated semiflow. The advantage of this approach is that the algebraic structure of a semiflow is simpler than the structure of a process. There is a price to pay, however: It turns out that the topological structure of the phase space of the associated semiflow is considerably more complex than that of the phase space of the process. Fortunately, many basic properties of semiflows do not rely on a specific topological structure; those pertaining to our work are collected, for convenience, in Section 3. They help to re-establish, in Section 5, the invariance principle for compact processes in strengthened form. More detailed information on oJ-limit sets is available for the class of uniform processes [3]. It is of interest to investigate whether these results can be re- obtained through a study of the associated semiflows. The difficulty is that equi- continuity and almost periodicity, which play a prominent role in the theory of * This research was supported by the Office of Naval Research under Contract No. N00014-67-A-0191-0020. 142 MATHEMATICAL SYSTEMS THEORY. Vol. 8, No. 2. © 1974 by Springer-Verlag New York Inc.