The Envelope Attractor of Non-strict Multivalued Dynamical Systems with Application to the 3D Navier–Stokes and Reaction–Diffusion Equations

Abstract

Multivalued semiflows generated by evolution equations without uniqueness sometimes satisfy a semigroup set inclusion rather than equality because, for example, the concatentation of solutions satisfying an energy inequality almost everywhere may not satisfy the energy inequality at the joining time. Such multivalued semiflows are said to be non-strict and their attractors need only be negatively semi-invariant. In this paper the problem of enveloping a non-strict multivalued dynamical system in a strict one is analyzed and their attactors are compared. Two constructions are proposed. In the first, the attainability set mapping is extending successively to be strict at the dyadic numbers, which essentially means (in the case of the Navier-Stokes system) that the energy inequality is satisfied piecewise on successively finer dyadic subintervals. The other deals directly with trajectories and their concatenations, which are then used to define a strict multivalued dynamical system. The first is shown to be applicable to the three-dimensional Navier-Stokes equations and the second to a reaction-diffusion problem without unique solutions.

Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations

“…On the other hand, since P is controllable, we take a controlled function φ such that it controls all elements in P t . Hence, by (11), the function φ controls all elements in ∪ s≤t Q s . Therefore, Q is backward controllable and so Q is backward T -compact.…”

“…In Sec.3, we discuss the backward topological property of a pullback attractor in the original spaces E and E 0 . The concept of a pullback attractor we use here depends on the trajectory space and the translation operator, which is slightly different from the stand one defined by an evolution process (see [1,4,8,9,12,13,25]), or a multi-valued process (see [5,6,10,11,26,30]), but they have some connections.…”

“…This phenomenon is similar to the 3D Navier-Stokes equation, the uniqueness of weak solutions remains open (cf. [11,19,21,22]). However, those weak solutions will generate a trajectory space, which consists of possibly multi-valued trajectories.…”