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.

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