## Counterexample in the plane for exterior domain

On page 75 of one Amick’s paper said,

Indeed, one can easily construct solenoidal velocity fields in $\Omega$ with finite Dirichlet norm which become unbounded like $(\ln(r))^\alpha$ with $0<\alpha<\frac{1}{2}$ as $r\to \infty$.

Actually the example here can be

$\psi=x\ln ^\alpha r$

$w=curl\psi=(\partial_y\psi,-\partial_x\psi)$

Apprently $w$ is a divergence free vector field. One can verify that the Dirichlet norm $\int_{|x|>1}|\nabla w|^2dx<\infty$ exactly when $\alpha<\frac{1}{2}$. Howver, $w$ grows like $\ln^\alpha r$ at infinity.

Remark: Charles J. Amick, On Leray’s problem of steady Navier-Stokes flow past a body in the plane.