The vanishing diffusion limit for an Oldroyd-B model in $\mathbb{R}^2_+$

We consider the initial-boundary value problem for an incompressible Oldroyd-B model with stress diffusion in two-dimensional upper half plane which describes the motion of viscoelastic polymeric fluids. From the physical point of view, the diffusive coefficient is several orders of magnitude smaller than other parameters in the model, and is usually assumed to be zero. However, the link between the diffusive model and the standard one (zero diffusion) via vanishing diffusion limit is still unknown from the mathematical point of view, in particular for the problem with boundary. Some numerical results [13] suggest that this should be true. In this work, we provide a rigorous justification for the vanishing diffusion in $L^\infty$-norm.