Related papers: Pointed vortex loops in ideal 2D fluids
This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called {\it the flow of finite type}, are in one-to-one correspondence with discrete structures such as…
Superfluid phase transitions are discussed from a geometrical perspective as envisaged by Onsager. The approach focuses on vortex loops which close to the critical temperature form a fluctuating vortex tangle. As the transition is…
In this work, we analytically study the existence of periodic vortex cap solutions for the homogeneous and incompressible Euler equations on the rotating unit 2-sphere, which was numerically conjectured by Dritschel-Polvani and…
We prove a gluing theorem for a symplectic vortex on a compact complex curve and a collection of holomorphic sphere bubbles. Using the theorem we show that the moduli space of regular stable symplectic vortices on a fixed curve with varying…
We introduce several new models whose common feature is to take into account effects from topological vorticity. The macroscopic unknown is driven by a dissipative anomalous diffusion (of SQG-type) and is coupled with the orientation of the…
At high densities fluids of strongly dipolar spherical particles exhibit spontaneous long-ranged orientational order. Typically, due to demagnetization effects induced by the long range of the dipolar interactions, the magnetization…
We present a finite element variational integrator for compressible flows. The numerical scheme is derived by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the…
In this paper, we consider the finite-time blowup of hollow vortices. These are solutions of the two-dimensional Euler equations for which the fluid domain is the complement of finitely many Jordan curves $\Gamma_1, \ldots, \Gamma_M$, and…
The Euler equation of an ideal (i.e. inviscid incompressible) fluid can be regarded, following V.Arnold, as the geodesic flow of the right-invariant $L^2$-metric on the group of volume-preserving diffeomorphisms of the flow domain. In this…
We construct two kinds of group cocycles on the volume-preserving diffeomorphism group. We show that, for the volume-preserving diffeomorphism group of the sphere, one of the cocycles gives the Euler class of flat sphere bundles.
We discuss the general properties of periodic vortex arrangements in rotating superfluids. The different possible structures are classified according to the symmetry space-groups and the circulation number. We calculate numerically several…
This article presents a comprehensive analysis of the formation and dissipation of vortices within chaotic fluid flows, leveraging the framework of Sobolev and Besov spaces on Riemannian manifolds. Building upon the Navier-Stokes equations,…
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…
Rotating, turbulent cosmic fluids are generally pervaded by coherent structures such as vortices and magnetic flux tubes. The formation of such structures is a robust property of rotating turbulence as has been confirmed in computer…
We discuss the configurations in which singly and doubly quantized vortex lines may coexist in a rotating superfluid. General principles of energy minimization lead to the conclusion that in equilibrium the two vortex species segregate…
We present the Hamiltonian formalism for the Euler equation of symplectic fluids, introduce symplectic vorticity, and study related invariants. In particular, this allows one to extend D.Ebin's long-time existence result for geodesics on…
We introduce a 2D network built of $\mathcal{PT}$-symmetric dimers with on-site cubic nonlinearity, the gain and loss elements of the dimers being linked by parallel square-shaped lattices. The system may be realized as a set of…
Concentration-compactness is used to prove compactness of maximising sequences for a variational problem governing symmetric steady vortex-pairs in a uniform planar ideal fluid flow, where the kinetic energy is to be maximised and the…
The aim of these notes is to present in a comprehensive and relatively self-contained way some recent developments in the mathematical analysis of two-dimensional viscous flows. We consider the incompressible Navier-Stokes equations in the…
We consider a wide class of approximate models of evolution of singular distributions of vorticity in three dimensional incompressible fluids and we show that they have global smooth solutions. The proof exploits the existence of suitable…