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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 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 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 c
By applying implicit function theorem on contour dynamics, we prove the existence of co-rotating and travelling patch solutions for both Euler and the generalized surface quasi-geostrophic equation. The solutions obtained constitute a desingularizati
In this paper, we obtain a uniform $W^{2,varepsilon}$-estimate of solutions to the fully nonlinear uniformly elliptic equations on Riemannian manifolds with a lower bound of sectional curvature using the ABP method.
The general stability problem of truncations for a family of functions concentrating mass at the origin is described and a concrete example in the framework of entire optimizers for the fractional Hardy-Sobolev inequality is given. In this short note