Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations.

*(English)*Zbl 0903.58020
J. Nonlinear Sci. 7, No. 5, 475-502 (1997); erratum ibid. 8, No. 2, 233 (1998).

A class of semiflows (generalized semiflows) with possibly nonunique solutions is considered. A generalized semiflow is defined to be a family of maps \(\varphi : [0,\infty)\to X\), where \(X\) is a metric space, satisfying axioms relating to existence, time translation, concatenation, and upper-semicontinuity with respect to initial data. It is shown that, under a mild technical hypothesis, for generalized semiflows strong measurability of solutions with respect to time implies their continuity on \((0,\infty)\). A generalized semiflow is shown to have a global attractor if and only if it is point dissipative and asymptotically compact. This result generalizes those for semiflows of J. K. Hale and O. A. Ladyzhenskaya. The structure of the global attractor in the presence of a Lyapunov function and its connectedness and stability properties are also studied. In particular, two examples (one finite- and the other infinite-dimensional) are given in which the global attractor is a single point but is not Lyapunov stable. Conditions under which the global attractor of a generalized semiflow is stable are also obtained. The theory of generalized semiflows is applied to the study of the 3D incompressible Navier-Stokes equations. It is shown that weak solutions satisfying an energy inequality form a generalized semiflow (in the phase space \(H\) consisting of \(L^2\) vector-fields with zero divergence) if and only if all weak solutions are continuous from \((0,\infty)\) to \(L^2\). Under the (unproved) hypothesis that all weak solutions are continuous from \((0,\infty)\) to \(L^2\), the existence of a global attractor is established.

Reviewer: E.Ershov (St.Peterburg)

##### MSC:

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |

37C10 | Dynamics induced by flows and semiflows |

35Q30 | Navier-Stokes equations |

54H20 | Topological dynamics (MSC2010) |