Related papers: Existence of Vortex Patch Equilibria for Active Sc…
The Sadovskii vortex patch is a traveling wave for the two-dimensional incompressible Euler equations consisting of an odd symmetric pair of vortex patches touching the symmetry axis. Its existence was first suggested by numerical…
This article investigates the dynamical behaviours of the $n$-vortex problem with vorticity $\mathbf{\Gamma}$ on a Riemann sphere $\mathbb{S}^2$ equipped with an arbitrary metric $g$. From perspectives of Riemannian geometry and symplectic…
We consider the $N$-vortex problem on the sphere assuming that all vortices have equal strength. We develop a theoretical framework to analyse solutions of the equations of motion with prescribed symmetries. Our construction relies on the…
Let $G=(V,E)$ be a connected finite graph. We study a system of non-Abelian multiple vortex equations on $G$. We established a necessary and sufficient condition for the existence and uniqueness of solutions to the non-Abelian multiple…
We derive regularized contour dynamics equations for the motion of infinite sharp fronts in the two-dimensional incompressible Euler, surface quasi-geostrophic (SQG), and generalized surface quasi-geostrophic (gSQG) equations. We derive a…
We consider the $N$-vortex problem on the sphere assuming that all vorticities have equal strength. We investigate relative equilibria (RE) consisting of $n$ latitudinal rings which are uniformly rotating about the vertical axis with…
In this paper, we prove nonlinear orbital stability for steady vortex patches that maximize the kinetic energy among isovortical rearrangements in a planar bounded domain. As a result, nonlinear stability for an isolated vortex patch is…
In this work, we analytically study the existence of periodic vortex cap solutions for the homogeneous and incompressible Euler equations on the rotating unit 2-sphere, which was numerically conjectured by Dritschel-Polvani and…
In this paper, we construct time periodic doubly connected solutions for the 3D quasi-geostrophic model in the patch setting. More specifically, we prove the existence of nontrivial $m$-fold doubly connected rotating patches bifurcating…
In this paper, we investigate the asymptotic behavior of solutions to the Navier-Stokes equations in the half-plane under high Reynolds number conditions, where the initial vorticity belongs to the Yudovich class and is supported away from…
It is well known that the incompressible Euler equations in two dimensions have globally regular solutions. The inviscid surface quasi-geostrophic (SQG) equation has a Biot-Savart law which is one derivative less regular than in the Euler…
This paper studies the dissipative generalized surface quasi-geostrophic equations in a supercritical regime where the order of the dissipation is small relative to order of the velocity, and the velocities are less regular than the…
It is well known that the Euler vortex patch in $\mathbb{R}^{2}$ will remain regular if it is regular enough initially. In bounded domains, the regularity theory for patch solutions is less complete. In this paper, we study Euler vortex…
We give an alternative proof of the nonuniqueness of weak solutions to the surface quasigeostrophic equation (SQG) first shown in [Buckmaster-Shkoller-Vicol, '16]. Our approach proceeds directly at the level of the scalar field.…
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…
In a companion paper, we gave a detailed account of the well-posedness theory for singular vortex patches. Here, we discuss the long-time dynamics of some of the classes of vortex patches we showed to be globally well-posed in the companion…
Vortices produce locally concentrated field configurations and are solutions to the nonlinear partial differential equations systems of complicated structures. In this paper, we establish the existence and uniqueness for solutions of the…
For the generalized surface quasi-geostrophic equation $$\left\{ \begin{aligned} & \partial_t \theta+u\cdot \nabla \theta=0, \quad \text{in } \mathbb{R}^2 \times (0,T), \\ & u=\nabla^\perp \psi, \quad \psi = (-\Delta)^{-s}\theta \quad…
Geophysical turbulent flows, characterized by rapid rotation, quantified by small Rossby number, and stable stratification, often self-organize into a collection of coherent vortices, referred to as a vortex gas. The lowest order asymptotic…
We constructed a family of steady vortex solutions for the lake equations with general vorticity function, which constitute a desingularization of a singular vortex. The precise localization of the asymptotic singular vortex is shown to be…