Journal of Mathematical Sciences, Vol. 94, No. 3, 1999 DYNAMICAL SYSTEMS GENERATED BY SOBOLEV CLASS VECTOR FIELDS IN FINITE AND INFINITE DIMENSIONS V. Bogaehev and E. Mayer-Wolf UDC 517.987.5 1. Introduction One of the fundamental problems of dynamical systems theory concerns the existence and uniqueness of a global flow {U~} generated by a vector field A. It is often desirable to have a measure/z which is invariant or quasi-invariant under {U~}. A standard result says that if A is uniformly Lipschitzian on ~ or on a Banach space, then there is a unique flow {Ut} generated by A. In addition, in the finite-dimensional case the Lebesgue measure is quasi-invariant under {Ut} (hence this is true for every measure with strictly positive density). However, in many applications, e.g., in fluid mechanics, statistical physics, stochastic analysis (especially in infinite dimensions) one has to deal with vector fields which are not even locally Lipschitzian, but still have certain regularity properties such as belonging to some Sobolev class. This kind of regularity does not always imply the continuity, so the very existence of solutions is not granted in advance. Recent infinite-dimensional analysis research, in particular in infinite-dimensional manifolds, gives rise to questions of the same sort (see, e.g., [24, 25; 30, 33, 36]). Even if we deal with uniformly bounded continuous fields on ]~'~, it Can happen that such a weak regularity is not enofigh, e.g., for uniqueness, An obvious example: n = 1, A(x) = 0 on (-c~, 0], A(x) = min(x ~, 1) on [0~c~),withc~ E (0,1). ThenAisin p,1 1 1 W/o r (]~) for p < -- but the corresponding equation has many 1--~' p,1 2 solutions. It is shown in [28] that for each p > 1, there is a field A E Cb(~ 2) f3 W/o c (~) generating infinitely many flows (that is, solutions with the group property) under which the Lebesgue measure is quasi-invariant. It also worth mentioning that in infinite dimensions the continuity of A does not imply even local solvability (see, e.g., [9, Sec. 2]). Before proceeding further, let us state with some precision what we mean by a solution to our dynamical systems. Definition 1.1. If X is a topological vector space equipped with a positive Radon measure # on its Borel ~-field and a measurable vector field A : X , X, a mapping (t, z) E ~ x X , uA(x) E X is said to be a solution to the equation t Ut(x) = x + / A(U,(x)) ds (1.1) 0 if (a) for/z-almost every x (1.1) holds with U = UA for all t E ~ (its right-hand side being implicitly well defined), and (b) for every t E ~, the measure/z o Ut -'1 is absolutely continuous with respect to #. The Radon-Nikodym derivative d (# o U~ 1) will always be denoted by rt. d# Moreover, U = U A is a flow solution if, in addition,/z-a.e. u,+, = u, o us vs,t ~ ~. (1.2) Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 42, Dinamicheskie Sistemy-6, 1997. 1394 1072-3374/99/9403-1394522.00 9 Kluwer Academic/Plenum Publishers