Symmetry in stationary and uniformly rotating solutions of active scalar equations

Abstract

In this paper, we study the radial symmetry properties of stationary and uniformlyrotating 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. In the patch setting, for the 2D Euler equation we show that every uniformly-rotating patch D with angular velocity Ω ≤ 0 or Ω ≥ 1 2 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 in Hmidi [48] and de la Hoz-Hassainia-Hmidi-Mateu [28] on uniformly-rotating patches. Along the way, we close a question on overdetermined problems for the fractional Laplacian [22, Remark 1.4], which may be of independent interest. The main new ideas come from a calculus of variations point of view.

Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

“…where U i is the domain enclosed by ∂ D i,in = i (thus U i is independent of ), and n is the outer normal of U i (thus the inner normal of D i ). The existence of c i is guaranteed by [14,Lemma 2.5]. One can then check that ∂U i u •ndσ = 0.…”

“…This type of rigidity question has been very lately understood for different equations and different settings such as in the papers by Koch-Nadirashvili-Sverak [20] for Navier-Stokes, Hamel-Nadirashvili [16][17][18] for the 2D Euler equation on a strip, punctured disk or the full plane, Gómez-Serrano-Park-Shi-Yao [14] for the 2D Euler and modified SQG in the full plane and Constantin-Drivas-Ginsberg [8] for the 2D and 3D Euler, as well as the 2D Boussinesq and the 3D Magnetohydrostatic (MHS) equations.…”

“…Structure of the proof. The proof is inspired by our recent rigidity result in the paper [14] on stationary and rotating solutions of the 2D Euler equations both in the smooth and vortex patch settings. To prove it, we constructed an appropriate functional and showed, on one hand, that any stationary solution had to be a critical point, and on the other, for any curve which is not a circle there existed a vector field along which the first variation was non-zero.…”

“…In [14] we proved a rearrangement inequality for such p in a similar spirit of Talenti's rearrangement inequality for elliptic equations [37], which we state below.…”

Symmetry results for compactly supported steady solutions of the 2D Euler equations

“…In [15] radial symmetry is proved for solutions in bounded domains under constant tangential velocity at the boundary: however this constant is not allowed to be 0. Another symmetry result is [10] for nonnegative and compactly supported vorticity, and here the velocity field need not have compact support (it is an immediate consequence of the divergence theorem that the unique compactly supported velocity field with nonnegative vorticity is 0).…”