The result is an equivalence, not a one-way implication
The paper's Theorem 1.1 states that, for a steady incompressible Euler flow in the exterior of a compact obstacle, two statements are equivalent: the velocity v(x) is nonzero for every x in the fluid domain, and the scalar vorticity ω is identically zero there. The boundary condition v·N > 0 on ∂K excludes zeros on the obstacle boundary, so any stagnation point must lie in the fluid domain Ω = R² \ K.
Reading this as an equivalence matters. It is not merely that irrotational flows happen to lack stagnation points; the converse also holds. If the vorticity is nonzero somewhere, the flow must contain a stagnation point. The authors describe this as a rigidity result: it restricts which boundary value problems can be solved in an exterior domain, because a flow without stagnation points is forced to be potential.
Three assumptions carry the whole theorem
The first assumption is the boundary condition v·N > 0 on ∂K, where N points from the obstacle K into the fluid Ω. Because the outward normal of Ω on ∂K is −N, this means fluid enters Ω through the boundary of the obstacle, and by continuity and compactness v·N ≥ c₀ > 0 for some c₀. The second and third assumptions concern infinity: there exist R₀, C > 0 such that v_r(x) = v(x)·e_r(x) > 0 and |v(x)|²/v_r(x) ≤ C/|x| for |x| ≥ R₀.
The radial condition says the distance from the origin increases along a trajectory once it lies outside a sufficiently large disk. The ratio condition controls circulation along such trajectories and implies |v(x)| ≤ C/|x|. A concrete sufficient condition is 0 < v_r ≤ M/|x| and |v_θ| ≤ K v_r outside a large disk, for some M, K > 0. The theorem does not require radial symmetry of the flow or of the obstacle, and it does not assume a positive uniform lower bound on the speed.
Irrotational flow has no stagnation points by the argument principle
For the direction from irrotational to no stagnation points, the proof uses the complex velocity q(z) = v₁(z) − i v₂(z). Incompressibility and zero vorticity give 2∂_z̄ q = div v − iω = 0, so q is holomorphic in Ω, and q(z) = 0 exactly when v(z) = 0.
The boundary condition fixes the winding number of q along ∂K at −1, and the positivity of v_r gives the same winding number −1 along every sufficiently large circle, both oriented counterclockwise. Applying the argument principle on the annulus between the obstacle and a large circle, the sum of the orders of the zeros of q equals −1 − (−1) = 0. Since every zero has positive multiplicity, q has no zeros there. This direction needs only the positivity of v_r at infinity, not the bound on |v|²/v_r.
No stagnation points force the vorticity to vanish
The converse direction is proved by contradiction. Assuming v never vanishes but ω(x*) ≠ 0 at some point x*, the authors first show that every trajectory escapes to infinity in forward time. In two dimensions the vorticity of a steady Euler flow is constant along trajectories, so a nonzero value can be propagated along a tube of trajectories on which the vorticity keeps a fixed sign and stays bounded away from zero.
The contradiction comes from estimating the integral of the vorticity over this truncated tube in two ways. The flux through each circular section is positive and independent of the radius, and combining this with |v(x)| ≤ C/|x| gives a quadratic lower bound for the area of the tube truncated at radius R, hence a quadratic lower bound for the absolute value of the vorticity integral. Stokes' theorem instead bounds that same integral using velocity values on the tube boundary, and the bound on |v|²/v_r makes the circulation around the tube grow at most logarithmically in R. Dividing by R² and letting R → ∞ forces δΦ_T/(2C) ≤ 0, contradicting δ > 0 and Φ_T > 0.
Velocity decay blocks the usual stream-function route
Earlier rigidity results for steady Euler flows often reduce the problem to a semilinear elliptic equation for a global stream function, or use the moving plane method. Those techniques are not directly available here. The positive flux through the obstacle boundary prevents a single-valued stream function from being defined on all of Ω, and the velocity decays at infinity rather than being bounded away from zero.
The paper also notes that the positive flux makes the energy infinite: with Φ = ∫_{∂K} v·N dℓ > 0, the divergence theorem on B_r \ K gives Φ = ∫_{∂B_r} v_r dℓ, and Cauchy's inequality yields ∫_{∂B_r} |v|² dℓ ≥ Φ²/(2πr), so ∫_Ω |v|² dx diverges. Results for finite-energy flows therefore do not apply in this setting. The authors also record that if the velocity is tangent to the obstacle boundary, a flow without stagnation points need not be irrotational; such circular examples have zero flux through every circle enclosing the obstacle and do not satisfy v_r > 0.
The same statement covers a magnetohydrostatic equilibrium
The problem can be rewritten in the language of ideal magnetohydrostatics. A static magnetic field B and plasma pressure p_m satisfying (curl B) × B = ∇p_m with div B = 0 become the steady Euler equations under the identification v = B and p = −p_m − ½|B|². For a two-dimensional field B = (B₁, B₂, 0), the scalar vorticity corresponds to the third component of curl B, and a stagnation point corresponds to a zero of the magnetic field.
Under the paper's boundary condition and assumptions at infinity, the theorem therefore implies that a two-dimensional equilibrium with nonzero current must have a point where the magnetic field vanishes. The authors connect this to the boundary value problems proposed by Grad and Rubin, where the normal component of the magnetic field is prescribed on the boundary together with additional data on the part through which the field enters. In this exterior problem the transport equation for the vorticity leads to a restriction rather than a construction: if the field does not vanish, the current must be zero.
What the equivalence rules in and out
The practical content of the theorem is a constraint on admissible exterior flows. If you are looking for a steady incompressible Euler flow in the exterior of a compact obstacle with fluid entering through every boundary point and with the stated decay at infinity, you cannot have both nonzero vorticity and a velocity that stays away from zero. Nonzero vorticity forces a stagnation point somewhere in Ω.
The authors flag the natural next question: identifying the weakest conditions at infinity under which the absence of stagnation points still forces the flow to be irrotational. That is the boundary of the present result. Within the stated assumptions, the sharpest way to read the theorem is as a dichotomy between two alternatives — either the flow is irrotational, or it has a stagnation point — with the boundary condition and the decay of the velocity at infinity doing the work that a uniform lower bound on the speed would do in bounded-domain rigidity results.
Sources used in this answer
Rigidity results for Euler flows in two dimensional exterior domains
Under the boundary condition v·N > 0 on ∂K and the assumptions v_r > 0 and |v|²/v_r ≤ C/|x| at infinity, a steady incompressible 2D Euler flow in the exterior of a compact obstacle is irrotational if and only if the velocity has no stagnation points.