On the flow the water under a still stone

Asymptotics of solutions of Stokes and Navier–Stokes systems is investigated at a point of tangency of smooth surfaces on which adhesion conditions are imposed. The leading terms of the asymptotics are described by special solutions of the resulting equation on the plane, which is similar to the Reynolds equation and generates at the origin. Complete asymptotic expansions are constructed (for any Reynolds number). Justification of these expansions is based on asymptotically exact estimates of the solutions in function spaces with weight norms. Solvability of the Navier–Stokes problem is established for small Reynolds numbers only.