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A simplified form of the vorticity equation is derived for arbitrary coordinate systems. The present work unifies and extends the previous findings that vorticity is conserved in planar Euler flow, while in axisymmetric Euler rings it is…
This paper addresses the long-time dynamics of solutions to the 2D incompressible Euler equations. We construct solutions with continuous vorticity $\omega_{\varepsilon}(x,t)$ concentrated around points $\xi_{j}(t)$ that converge to a sum…
A detailed study of complex-space singularities of the two-dimensional incompressible Euler equation is performed in the short-time asymptotic r\'egime when such singularities are very far from the real domain; this allows an exact…
We consider the limit $\alpha\to0$ for the $\alpha$-Euler equations in a two-dimensional bounded domain with Dirichlet boundary conditions. Assuming that the vorticity is bounded in $L^p$, we prove the existence of a global solution and we…
In the analysis of the long-time behaviour of two-dimensional incompressible viscous fluids, Oseen vortices play a major role as attractors of any homogeneous solution with integrable initial vorticity. As a first step in the study of the…
We consider the two-dimensional Euler equations in non-smooth domains with corners. It is shown that if the angle of the corner $\theta$ is strictly less than $\pi/2$, the Lipschitz estimate of the vorticity at the corner is at most single…
Coherent vortices are often observed to persist for long times in turbulent 2D flows even at very high Reynolds numbers and are observed in experiments and computer simulations to potentially be asymptotically stable in a weak sense for the…
We consider the incompressible Euler equations in $R^2$ when the initial vorticity is bounded, radially symmetric and non-increasing in the radial direction. Such a radial distribution is stationary, and we show that the monotonicity…
In this paper, we investigate steady Euler flows in a two-dimensional bounded domain. By an adaption of the vorticity method, we prove that for any nonconstant harmonic function $q$, which corresponds to a nontrivial irrotational flow,…
We prove the asymptotic stability of shear flows close to the Couette flow for the 2-D inhomogeneous incompressible Euler equations on $\mathbb{T}\times \mathbb{R}$. More precisely, if the initial velocity is close to the Couette flow and…
A second-order expansion of the Gross-Pitaevskii equation in the interaction parameter determines the thermodynamic critical angular velocity Omega_c for the creation of a vortex in a small axisymmetric condensate. Similarly, a second-order…
We are interested in the stability analysis of two-dimensional incompressible inviscid fluids. Specifically, we revisit a recent result on the stability of Yudovich's solutions to the incompressible Euler equations in $L^\infty([0,T];H^1)$…
We consider the incompressible Euler equations in the half cylinder $ \mathbb{R}_{>0}\times\mathbb{T}$. In this domain, any vorticity which is independent of $x_2$ defines a stationary solution. We prove that such a stationary solution is…
We study the asymptotic behavior and the asymptotic stability of the two-dimensional Euler equations and of the two-dimensional linearized Euler equations close to parallel flows. We focus on spectrally stable jet profiles $U(y)$ with…
We consider solutions to the Navier-Stokes equations on $\mathbb{R}^2$ close to the Poiseuille flow with viscosity $0< \nu < 1$. For the linearized problem, we prove that when the $x$-frequency satisfy $|k| \ge \nu^{-\frac{1}{3}}$, the…
We consider the 3D incompressible Euler equations under the following situation: small-scale vortex blob being stretched by a prescribed large-scale stationary flow. More precisely, we clarify what kind of large-scale stationary flows…
For the 2D incompressible Euler equations, we establish global-in-time ($t \in \mathbb{R}$) stability of vortex quadrupoles satisfying odd symmetry with respect to both axes. Specifically, if the vorticity restricted to a quadrant is…
We study the time evolution of an incompressible Euler fluid with planar symmetry when the vorticity is initially concentrated in small disks. We discuss how long this concentration persists, showing that in some cases this happens for…
The purpose of this work is to prove existence of a weak solution of the two dimensional incompressible Euler equations on a noncylindrical domain consisting of a smooth, bounded, connected and simply connected domain undergoing a…
For any $A > 2$, we construct solutions to the two-dimensional incompressible Euler equations on the torus $\mathbb{T}^2$ whose vorticity gradient $\nabla\omega$ grows exponentially in time: $$\|\nabla\omega(t, \cdot)\|_{L^\infty} \gtrsim…