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Register a new userRemarks on stationary and uniformly-rotating vortex sheets: Rigidity results
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In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $Omega$, such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.
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In this paper, we construct new, uniformly-rotating solutions of the vortex sheet equation bifurcating from circles with constant vorticity amplitude. The proof is accomplished via a Lyapunov-Schmidt reduction and a second order expansion of the reduced system.
In this paper, we derive some quantitative estimates for uniformly-rotating vortex patches. We prove that if a non-radial simply-connected patch $D$ is uniformly-rotating with small angular velocity $0 < Omega ll 1$, then the outmost point of the patch must be far from the center of rotation, with distance at least of order $Omega^{-1/2}$. For $m$-fold symmetric simply-connected rotating patches, we show that their angular velocity must be close to $frac{1}{2}$ for $mgg 1$ with the difference at most $O(1/m)$, and also obtain estimates on $L^{infty}$ norm of the polar graph which parametrizes the boundary.
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 $Omega leq 0$ or $Omegageq frac{1}{2}$ must be radial, where both bounds are sharp. For the gSQG equation we obtain a similar symmetry result for $Omegaleq 0$ or $Omegageq Omega_alpha$ (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.
We investigate a steady planar flow of an ideal fluid in a (bounded or unbounded) domain $Omegasubset mathbb{R}^2$. Let $kappa_i ot=0$, $i=1,ldots, m$, be $m$ arbitrary fixed constants. For any given non-degenerate critical point $mathbf{x}_0=(x_{0,1},ldots,x_{0,m})$ of the Kirchhoff-Routh function defined on $Omega^m$ corresponding to $(kappa_1,ldots, kappa_m)$, we construct a family of stationary planar flows with vortex sheets that have large vorticity amplitude and are perturbations of small circles centered near $x_i$, $i=1,ldots,m$. The proof is accomplished via the implicit function theorem with suitable choice of function spaces. This seems to be the first nontrivial result on the existence of stationary vortex sheets in domains.
We construct co-rotating and traveling vortex sheets for 2D incompressible Euler equation, which are supported on several small closed curves. These examples represent a new type of vortex sheet solutions other than two known classes. The construction is based on Birkhoff-Rott operator, and accomplished by using implicit function theorem at point vortex solutions with suitably chosen function spaces.
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