Symmetry in stationary and uniformly rotating solutions of active scalar equations

We study the radial symmetry properties of stationary and uniformly rotating solutions of the 2D Euler and gSQG equations, both in the smooth setting and the patch setting. For the 2D Euler equation, we show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial, without any assumptions on the connectedness of the support or the level sets. For the 2D Euler equation in the patch setting, we show that every uniformly rotating patch D with angular velocity Ω ≤ 0 or Ω ≥ ¹₂ must be radial, where both bounds are sharp. For the gSQG equation, we obtain a similar symmetry result for Ω ≤ 0 or Ω ≥ Ωα (with the bounds being sharp), under the additional assumption that the patch is simply connected. These results settle several open questions posed by Hmidi, de la Hoz, Hassainia, and Mateu on uniformly rotating patches. Along the way, we close a question by Choksi, Neumayer, and Topaloglu on overdetermined problems for the fractional Laplacian, which may be of independent interest. The main new ideas come from a calculus-of-variations point of view.