Two-dimensional flow around an obstacle at large Reynolds numbers

Establish the existence and rigorous solvability theory for the two-dimensional steady Navier–Stokes flow around an obstacle at large Reynolds numbers, subject to no-slip boundary conditions on the obstacle and convergence to a prescribed uniform velocity at infinity.

Background

The paper considers finite-Dirichlet-integral solutions of the steady two-dimensional Navier–Stokes equations in possibly unbounded plane domains. The authors identify the exterior flow-around-an-obstacle problem as a major unresolved issue in mathematical hydrodynamics.

The difficulty is particularly pronounced in two dimensions because finite Dirichlet integral does not imply boundedness or suitable decay of the velocity at infinity, reflecting the failure of the corresponding three-dimensional Sobolev embedding and the associated Stokes paradox.

References

Still there are a lot of open problems here, for example, the famous flow around an obstacle problem, solved for 3d case almost a century ago, but still open for 2d case for big Reynolds numbers (see, e.g., (\ref{NSobs}) for the formulation and for a recent survey).

— Sharp Basic Velocity Estimates for the Plane Steady Navier--Stokes Equations through Vorticity  (2609.30844 - Korobkov et al., 25 Sep 2026) in Section 1, Introduction