Continuity of pullback random attractors for three-dimensional stochastic globally modified Navier–Stokes equations

Abstract We study the full continuity in expectation of pullback random attractors for the three-dimensional globally modified Navier–Stokes equations driven by multiplicative white noise and time-dependent forces. By proving the local compactness of the pullback random attractors <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi mathvariant="script">A</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>τ</m:mi> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msub> <m:mi>ω</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:mspace width="0.17em"/> <m:mi>τ</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> $\left\{\mathcal{A}\left(\tau ,{\theta }_{s}\omega \right) : \tau ,s\in \mathbb{R}\right\}$ , we establish their continuity in expectation with respect to the Hausdorff metric. As a consequence, two different kinds of continuities in ( τ , s ), including residual dense continuity and full upper semicontinuity are obtained. Due to the presence of the nonlinear term in the underlying system, the analysis of the continuity of pullback random attractors becomes more difficult and interesting.