Asymptotics of Laplace and Stokes solutions in the gap between a sharp tip and a smooth boundary

We derive an explicit description of the asymptotic singular behavior of 2D Laplace and Stokes solutions as the distance between a Lipschitz sharp tip and a smooth boundary vanishes. While classical lubrication theory for smoother boundaries relies on the anisotropy of the gap to perform dimensional reduction, the isotropic scaling imposed by a corner singularity requires a different approach. We introduce a stability-based method that allows us to construct the leading-order asymptotics of these solutions in the gap. As an application, we determine the asymptotic behavior of the Stokes resistance matrix and the pressure field.