The asymptotic analysis of two-dimensional Oldroyd fluid flow equations for viscoelastic fluids in a bounded domain \(\Omega \) with a \(\textrm{C}^2\) -boundary \(\partial \Omega ,\) is carried out in this work. If the forcing term \(\varvec{f}\) is in the space \(\mathbb {L}^2(\Omega )\) , then we show that the global attractor for 2D Oldroyd system is compact not only in the \(\mathbb {L}^2\) -norm but also in the \(\mathbb {H}^1\) -norm whenever the divergence-free initial data is in \(\mathbb {H}^1_{0}(\Omega )\) . The existence of an absorbing set and flattening property of the semigroup associated with the concerned system are exploited in the proofs.
