[Paper Review] Symmetry in stationary and uniformly-rotating solutions of active scalar equations

Javier Gómez-Serrano, Jaemin Park+2 more|arXiv (Cornell University)|Aug 5, 2019

Navier-Stokes equation solutions64 references19 citations

TL;DR

This paper establishes radial symmetry for stationary and uniformly-rotating solutions of the 2D Euler and generalized Surface Quasi-Geostrophic (gSQG) equations under sharp angular velocity bounds. Using a calculus of variations approach, it proves that smooth, compactly supported, nonnegative vorticity solutions to the 2D Euler equation are radially symmetric, and that uniformly-rotating patches for both 2D Euler and gSQG are radial when angular velocity Ω ≤ 0 or Ω ≥ 1/2 (for 2D Euler) or Ω ≥ Ωₐ (for gSQG), with sharp thresholds.

ABSTRACT

In this paper, 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. In the patch setting, for the 2D Euler equation we show that every uniformly-rotating patch $D$ with angular velocity $Ω\leq 0$ or $Ω\geq \frac{1}{2}$ must be radial, where both bounds are sharp. For the gSQG equation we obtain a similar symmetry result for $Ω\leq 0$ or $Ω\geq Ω_α$ (with the bounds being sharp), under the additional assumption that the patch is simply-connected. These results settle several open questions in [T. Hmidi, J. Evol. Equ., 15(4): 801-816, 2015] and [F. de la Hoz, Z. Hassainia, T. Hmidi, and J. Mateu, Anal. PDE, 9(7):1609-1670, 2016] on uniformly-rotating patches. Along the way, we close a question on overdetermined problems for the fractional Laplacian [R. Choksi, R. Neumayer, and I. Topaloglu, Arxiv preprint arXiv:1810.08304, 2018, Remark 1.4], which may be of independent interest. The main new ideas come from a calculus of variations point of view.

Motivation & Objective

• To resolve open questions on radial symmetry of uniformly-rotating patches in active scalar equations.
• To establish sharp bounds on angular velocity Ω for which uniformly-rotating patches must be radially symmetric.
• To close a question on overdetermined problems for the fractional Laplacian, which may have independent interest.
• To provide symmetry results for both smooth and patch solutions in 2D Euler and gSQG equations without assuming connectedness of the support.

Proposed method

• A calculus of variations framework is employed to analyze the symmetry properties of solutions.
• The analysis relies on integral equations derived from the Biot–Savart law: $1_D * K_\alpha - \frac{\Omega}{2}|x|^2 \equiv C_i$ on $\partial D$ for patches.
• For gSQG, the method assumes simply-connected patches and uses the kernel $K_\alpha(x) = -C_\alpha |x|^{-\alpha}$ for $\alpha \in (0,2)$.
• The proof uses comparison arguments and properties of the beta and gamma functions to derive sharp bounds on $\Omega$.
• Symmetry is established via contradiction and variational characterization, avoiding assumptions on level set connectivity.
• The analysis includes quantitative estimates on the symmetric difference between a rotating patch and a disk.

Experimental results

Research questions

• RQ1 Under what conditions must a uniformly-rotating patch in the 2D Euler or gSQG equation be radially symmetric?
• RQ2 Are there sharp thresholds for the angular velocity Ω beyond which symmetry is guaranteed?
• RQ3 Can radial symmetry be proven for smooth, compactly supported, nonnegative vorticity solutions without assuming connected support?
• RQ4 Does the symmetry result for patches extend to the gSQG equation with $\alpha \in (0,2)$, and what are the sharp bounds?
• RQ5 Can the overdetermined problem for the fractional Laplacian be resolved using this framework?

Key findings

• Any smooth, compactly supported, nonnegative vorticity solution to the 2D Euler equation must be radially symmetric, regardless of the connectedness of the support or level sets.
• For the 2D Euler equation, every uniformly-rotating patch with $\Omega \leq 0$ or $\Omega \geq \frac{1}{2}$ must be radial, and both bounds are sharp.
• For the gSQG equation with $\alpha \in (0,1)$, uniformly-rotating simply-connected patches with $\Omega \leq 0$ or $\Omega \geq \Omega_\alpha$ must be radial, and $\Omega_\alpha$ is the sharp threshold.
• The paper resolves open questions from Hmidi [48] and de la Hoz–Hassainia–Hmidi–Mateu [28] on uniformly-rotating patches.
• A quantitative estimate is obtained: if a simply-connected patch of area $\pi$ rotates with $\Omega \in (0, \Omega_\alpha)$, then its symmetric difference with the unit disk satisfies $|D \triangle B| \leq 2\pi\left(\left(\frac{\Omega_\alpha}{\Omega}\right)^{2/\alpha} - 1\right)$.
• The paper closes a question on overdetermined problems for the fractional Laplacian, as posed in [22, Remark 1.4], using the same variational framework.