相关论文: $C^1$ type regularization for point vortices on $\…
The goal of this numerical study is to get insight into singular solutions of the two-dimensional (2D) Euler equations for non-smooth initial data, in particular for vortex sheets. To this end high resolution computations of vortex layers…
It is an open question whether solutions of the Einstein-Euler equations are smooth enough to admit locally inertial coordinates at points of shock wave interaction, or whether "regularity singularities" can exist at such points. The term…
We study desingularization of steady vortex rings in three-dimensional axisymmetric incompressible Euler fluids with swirl. Using the variational method, we construct a two-parameter family of steady vortex rings, which constitute a…
In this paper, we are concerned with nonlinear desingularization of steady vortex rings of three-dimensional incompressible Euler fluids. We focus on the case when the vorticity function has a simple discontinuity, which corresponding to a…
We study a 1D fluid mechanics model with nonlocal velocity. The equation can be viewed as a fractional porous medium flow, a 1D model of the quasi-geostrophic equation, and also a special case of Euler-Alignment system. For strictly…
In this paper, we prove that in bounded planar domains with $C^{2,\alpha}$ boundary, for almost every initial condition in the sense of the Lebesgue measure, the point vortex system has a global solution, meaning that there is no collision…
In this paper we consider the incompressible 2D Euler equation in an annular domain with non-penetration boundary condition. In this setting, we prove the existence of a family of non-trivially smooth time-periodic solutions at an…
This article studies the N-vortex problem in the plane with positive vorticities. After an investigation of some properties for normalised relative equilibria of the system, we use symplectic capacity theory to show that, there exist…
For the first time the exact vortex solution of the Cauchy problem in unbounded space is obtained for the three-dimensional Euler-Helmholtz (EH) equation in the case of a nonzero-divergence velocity field for an ideal compressible medium.…
By establishing a sharp Strichartz estimate for the velocity and density, we prove the local well-posedness of solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial velocity, density, and…
We study vortex-like solutions to the Lifshitz-Chern-Simons theory. We find that such solutions exists and have a logarithmically divergent energy, which suggests that a Kostelitz-Thouless transition may occur, in which voxtex-antivortex…
We establish the local existence and uniqueness of solutions to the two-dimensional compressible Euler equations with initial velocity $\bv_0$, logarithmic density $\rho_0$, and specific vorticity \(w_0\), which satisfy $(\bv_0, \rho_0,…
We study the minimizer u of a convex functional in the plane which is not G\^ateaux-differentiable. Namely, we show that the set of critical points of any C^1-smooth minimizer can not have isolated points. Also, by means of some appropriate…
We study the dynamics of 3 point-vortices on the plane for a fluid governed by Euler's equations, concentrating on the case when the moment of inertia is zero. We prove that the only motions that lead to total collisions are self-similar…
We present a new application of Lagrangian Perturbation Theory (LPT): the stability analysis of fluid flows. As a test case that demonstrates the framework we focus on the plane Couette flow. The incompressible Navier-Stokes equation is…
A class of axisymmetric vortex solutions superposed upon radial stagnation flows is described. The new vortex solutions generalize the classical Burgers' vortex and Sullivan's vortex solutions in the presence of a volumetric line source at…
This article revisits the instability of sharp shear interfaces, also called vortex sheets, in incompressible fluid flows. We study the Birkhoff-Rott equation, which describes the motion of vortex sheets according to the incompressible…
We rigorously establish the formal asymptotics of Neu for Gross-Pitaevskii vortex dynamics in the plane. Given any integer $n\geq2$, we construct a family of $n$-vortex solutions with vortices of degree $\pm1$, and describe precisely the…
This paper deals with the existence of $N$ vortex patches located at the vertex of a regular polygon with $N$ sides that rotate around the center of the polygon at a constant angular velocity. That is done for Euler and (SQG)$_\beta$…
The vortex method is a common numerical and theoretical approach used to implement the motion of an ideal flow, in which the vorticity is approximated by a sum of point vortices, so that the Euler equations read as a system of ordinary…