Stokes' paradox

Counterexample

Let $D = \set{x \in \R^2 : \abs{x} \le a}$ be a disk of radius $a > 0$, the cross-section of an infinite circular cylinder, and let $U \ne 0$ be a speed. Consider the two-dimensional steady Stokes equations $\mu \Delta u = \nabla p$, $\nabla \cdot u = 0$ in $\R^2 \setminus D$. They have no smooth solution $(u, p)$ with $$u = 0 \text{ on } \abs{x} = a \qquad \text{and} \qquad u(x) \to (U, 0) \text{ as } \abs{x} \to \infty.$$ So slow uniform flow past a cylinder has no Stokes solution in the plane. For a sphere in three dimensions, the same problem is solvable and gives Stokes' law.

Why: a body that feels a nonzero force $F$ produces at large distances a two-dimensional Stokeslet, whose velocity grows like $\abs{F} \log \abs{x}$. One cannot match such a flow both to the no-slip condition on the cylinder and to a finite uniform stream at infinity.