Related papers: Axi-symmetrization near point vortex solutions for…
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…
We consider concentrated vorticities for the Euler equation on a smooth domain $\Omega \subset \mathbf{R}^2$ in the form of \[ \omega = \sum_{j=1}^N \omega_j \chi_{\Omega_j}, \quad |\Omega_j| = \pi r_j^2, \quad \int_{\Omega_j} \omega_j d\mu…
We rigorously construct the first steady traveling wave solutions of the 2D incompressible Euler equation that take the form of a contiguous vortex-patch dipole, which can be viewed as the vortex-patch counterpart of the well-known…
We prove that the 2d Euler equation is globally well-posed in a space of vector fields having spatial asymptotic expansion at infinity of any a priori given order. The asymptotic coefficients of the solutions are holomorphic functions of…
It is well-known that the dynamics of vortices in an ideal incompressible two-dimensional fluid contained in a bounded not necessarily simply connected smooth domain is described by the Kirchhoff--Routh point vortex system. In this paper,…
In this paper, we construct a family of global solutions to the incompressible Euler equation on a standard 2-sphere. These solutions are odd-symmetric with respect to the equatorial plane and rotate with a constant angular speed around the…
A {\em vortex pair} solution of the incompressible $2d$ Euler equation in vorticity form $$ \omega_t + \nabla^\perp \Psi\cdot \nabla \omega = 0 , \quad \Psi = (-\Delta)^{-1} \omega, \quad \hbox{in } \mathbb{R}^2 \times (0,\infty)$$ is a…
We study weak solutions of the two-dimensional (2D) filtered Euler equations whose vorticity is a finite Radon measure and velocity has locally finite kinetic energy, which is called the vortex sheet solution. The filtered Euler equations…
In this paper, we study desingularization of vortices for the two-dimensional incompressible Euler equations in the full plane. We construct a family of steady vortex pairs for the Euler equations with a general vorticity function, which…
This paper addresses the construction and the stability of self-similar solutions to the isentropic compressible Euler equations. These solutions model a gas that implodes isotropically, ending in a singularity formation in finite time. The…
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 prove the existence of a unique global strong solution for a stochastic two-dimensional Euler vorticity equation for incompressible flows with noise of transport type. In particular, we show that the initial smoothness of the solution is…
We show that, in a two-dimensional (2d) ideal fluid (also applies to a column of quasi-2d non-neutral plasma in an axial magnetic field), large elliptical vortices in a finite disk are stable. The stability is established by comparison…
This paper concerns the stabilizing effect of viscosity on the vortex sheets. It is found that although a vortex sheet is not a time-asymptotic attractor for the compressible Navier-Stokes equations, a viscous wave that approximates the…
We consider relative equilibrium solutions of the two-dimensional Euler equations in which the vorticity is concentrated on a union of finite-length vortex sheets. Using methods of complex analysis, more specifically the theory of the…
Let $v$ be a solution of the axially symmetric Euler equations (ASE) in a finite cylinder in $\mathbb{R}^3$. We show that suitable blow-up limits of possible velocity singularity and most self similar vorticity singularity near maximal…
We construct a new class of self-similar implosion profiles for the multi-dimensional compressible Euler equations. These profiles are smooth, genuinely non-isentropic, radially/spherically symmetric, and have explicit (closed-form)…
Consider a $1$D simple small-amplitude solution $(\rho_{(bkg)}, v^1_{(bkg)})$ to the isentropic compressible Euler equations which has smooth initial data, coincides with a constant state outside a compact set, and forms a shock in finite…
In this paper, we study the stability two-dimensional (2D) steady Euler flows with sharply concentrated vorticity in a simply-connected bounded domain. These flows are obtained as maximizers of the kinetic energy subject to the constraint…
We review and apply the continuous symmetry approach to find the solution of the 3D Euler fluid equations in several instances of interest, via the construction of constants of motion and infinitesimal symmetries, without recourse to…