Related papers: Vortex Filament Equation for a regular polygon in …
In this paper, we study the evolution of the vortex filament equation (VFE), $$\mathbf X_t = \mathbf X_s \wedge \mathbf X_{ss},$$ with $\mathbf X(s, 0)$ being a regular planar polygon. Using algebraic techniques, supported by full numerical…
In this paper, we consider the evolution of the Vortex Filament equation (VFE): \begin{equation*} \mathbf X_t = \mathbf Xs \wedge \mathbf Xss, \end{equation*} taking $M$-sided regular polygons with nonzero torsion as initial data. Using…
In this paper, we give evidence that the evolution of the Vortex Filament Equation for a regular $M$-corner polygon as initial datum can be explained at infinitesimal times as the superposition of $M$ one-corner initial data. Therefore, and…
In this paper, we consider the evolution of the so-called vortex filament equation (VFE), \begin{equation*} \mathbf X_t = \mathbf X_s\wedge\mathbf X_{ss}, \end{equation*} taking a planar regular polygon of $M$ sides as initial datum. We…
For the class of quasi-periodic solutions of the vortex filament equation, we study connections between the algebro-geometric data used for their explicit construction and the geometry of the evolving curves. We give a complete description…
In this proceedings article we shall survey a series of results on the stability of self-similar solutions of the vortex filament equation. This equation is a geometric flow for curves in $\mathbb R^3$ and it is used as a model for the…
The evolution of a vortex line following the binormal flow equation (i.e. with a velocity proportional to the local curvature in the direction of the binormal vector) has been postulated as an approximation for the evolution of vortex…
In this paper, we study the evolution of a vortex filament in an incompressible ideal fluid. Under the assumption that the vorticity is concentrated along a smooth curve in $\mathbb{R}^3$, we prove that the curve evolves to leading order by…
Klein, Majda, and Damodaran have previously developed a formalized asymptotic motion law describing the evolution of nearly parallel vortex filaments within the framework of the three-dimensional Euler equations for incompressible fluids.…
In this paper we present an error analysis of an Eulerian finite element method for solving parabolic partial differential equations posed on evolving hypersurfaces in $\mathbb{R}^d$, $d=2,3$. The method employs discontinuous piecewise…
We study an evolution problem in the space of continuous loops in three-dimensional Euclidean space modelled upon the dynamics of vortex lines in 3d incompressible and inviscid fluids. We establish existence of a local solution starting…
In this paper, we give a rigorous proof for the expression of the angle between adjacent sides in the skew polygons appearing at rational times in the evolution of regular polygons of $M$ sides under the vortex filament equation. The proof…
The motion of a vortex filament in superfluid 4He is considered by using the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) phenomenological model for the scattering process between the vortex and thermal excitations in liquid 4He. The HVBK…
We prove that for solutions of the Euler equation on the sphere, the vorticity gradient can grow at most double-exponentially in time, and we show that this upper bound is sharp by constructing explicit solutions with odd symmetry that…
We describe ideal incompressible hydrodynamics on the hyperbolic plane which is an infinite surface of constant negative curvature. We derive equations of motion, general symmetries and conservation laws, and then consider turbulence with…
We consider a nonlinear model equation, known as the Localized Induction Equation, describing the motion of a vortex filament immersed in an incompressible and inviscid fluid. We prove the unique solvability of an initial-boundary value…
The $n+1$ vortex filament problem has explicit solutions consisting of $n$ parallel filaments of equal circulation in the form of nested polygons uniformly rotating around a central filament which has circulation of opposite sign. We show…
This paper gives an analysis of the movement of n+1 almost parallel filaments or vortices. Starting from a polygonal equilibrium of n vortices with equal circulation and one vortex at the center of the polygon, we find bifurcation of…
A class of harmonic solutions to the steady Euler equations for incompressible fluids is presented in two dimensions in circular, elliptic and bipolar coordinates. Since the velocity field is solenoidal in this case, it can be written as…
A model equation for the motion of a vortex filament immersed in three dimensional, incompressible and inviscid fluid is investigated as a humble attempt to model the motion of a tornado. We solve an initial-boundary value problem in the…