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Backward stability and divided invariance of an attractor for the delayed Navier-Stokes equation. (English) Zbl 1447.35072
Summary: We study backward stability of a pullback attractor especially for a delay equation. We introduce a new concept of a backward attractor, which is defined by a compact, pullback attracting and dividedly invariant family. We then show the equivalence between existence of a backward attractor and backward stability of the pullback attractor, and present some criteria by using the backward limit-set compactness of the system. In the application part, we consider the Navier-Stokes equation with a nonuniform Lipschitz delay term and a backward tempered force. Based on the fact that the delay does not change the backward bounds of the velocity field and external forces, we establish the backward-uniform estimates and obtain a backward attractor, which leads to backward stability of the pullback attractor. Some special cases of variable delay and distributed delay are discussed.

##### MSC:
 35B41 Attractors 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 35Q30 Navier-Stokes equations
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