Weakest far-field conditions forcing irrotationality

Identify the weakest conditions at infinity for a two-dimensional steady incompressible Euler flow in the exterior of a bounded obstacle under which the absence of stagnation points forces the flow to be irrotational.

Background

The paper proves that, for steady two-dimensional incompressible Euler flows in the exterior of a bounded obstacle, positive normal velocity through the obstacle boundary together with the far-field assumptions v_r>0 and |v|^2/v_r\leq C/|x| imply an equivalence between the absence of stagnation points and vanishing vorticity. The authors note that the specific decay and directional hypotheses at infinity are sufficient for their argument but may not be optimal.

The unresolved problem is to characterize the weakest asymptotic conditions at infinity that still guarantee this rigidity conclusion: namely, that a velocity field with no zeros must be irrotational. This would generalize the main theorem by reducing or replacing the current far-field assumptions.