Vortex Filament Solutions of the Navier‐Stokes Equations

Finite-Gap Solutions of the Vortex Filament Equation: Isoperiodic Deformations

We study the topology of quasiperiodic solutions of the vortex filament equation in a neighborhood of multiply covered circles. We construct these solutions by means of a sequence of isoperiodic deformations, at each step of which a real double point is “unpinched” to produce a new pair of branch points and therefore a solution of higher genus. We prove that every step in this process correspon...

Global Solutions of the Random Vortex Filament Equation

In this article we prove the existence of a global solution for the random vortex filament equation. Our work gives a positive answer to a question left open in recent publications: Berselli and Gubinelli [5] showed the existence of global solution for a smooth initial condition while Bessaih, Gubinelli, Russo [6] proved the existence of a local solution for a general initial condition. In this...

Finite-Gap Solutions of the Vortex Filament Equation: Genus One Solutions and Symmetric Solutions

For the class of quasiperiodic 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 of genus one solutions, including geometrically interesting special cases such as Euler elastica, constant torsion curves, and self-intersecting filaments. W...

Vortex solutions of the evolutionary Ginzburg-Landau type equations

We consider two types of the time-dependent Ginzburg-Landau equation in 2D bounded domains: the heat-flow equation and the Schrödinger equation. We study the asymptotic behaviour of the vortex solutions of these equations when the vortex core size is much smaller than the inter-vortex distance. Using the method of the asymptotic expansion near the vortices, we obtain the systems of ordinary dif...

Numerical Simulation of Vortex Breakdown by the Vortex-Filament Method