Pullback attractors for the non-autonomous 2D Navier-Stokes equations for minimally regular forcing

Abstract

This paper treats the existence of pullback attractors for the non-autonomous 2D Navier--Stokes equations in two different spaces, namely L^2 and H^1. The non-autonomous forcing term is taken in L^2_{\rm loc}(\mathbb R;H^{-1}) and L^2_{\rm loc}(\mathbb R;L^2) respectively for these two results: even in the autonomous case it is not straightforward to show the required asymptotic compactness of the flow with this regularity of the forcing term. Here we prove the asymptotic compactness of the corresponding processes by verifying the flattening property -- also known as Condition (C)". We also show, using the semigroup method, that a little additional regularity -- f\in L^p_{\rm loc}(\mathbb R;H^{-1}) or f\in L^p_{\rm loc}(\mathbb R;L^2) for some p>2 -- is enough to ensure the existence of a compact pullback absorbing family (not only asymptotic compactness). Even in the autonomous case the existence of a compact absorbing set for this model is new when f has such limited regularity.