51 problems

Vortex filament conjecture. Vorticity initially concentrated around a curve should remain close to a curve evolving according to the binormal-curvature equation above, to leading o…

Let the $N$-vortex moduli space be the space of critically coupled $N$-vortex solutions, equipped with the asymptotic Kähler form … Here $Z$ is the centre-of-mass coordinate,…

A helical vortex tube is a vortex structure invariant under a helical symmetry, and a three-dimensional incompressible Euler flow is a velocity field satisfying the incompressible…

Vortex filament conjecture. For such initial data, the corresponding solution $w(x,t)$ remains concentrated near

Vortex filament conjecture. For such initial data, the corresponding solution remains concentrated near a point moving according to

Gallay–Maekawa's conjecture. As $|\alpha|\to\infty$,

Small-viscosity approximation conjecture. For sufficiently small $u$, the dynamical behavior of the solutions to the viscous equation should sufficiently approximate that of the c…

Let $T_\rho$ denote the time of validity of the evolution result for helical vortices with general initial data, and let $T^*$ be the corresponding limiting time in the vortex-ring…

Let $\V_b^{1}[]$ be the first component of the velocity at the origin for parameters $\beta\in(0,1)$, $\b\in(0,1)$, and outer radius $\R>1$. Based on numerical experiments, positiv…

Let $\boldsymbol{\phi}_1$ and $\boldsymbol{\phi}_{1,b}$ denote the functions appearing in the definition of $V_b^{2}[]$, with $b\in(0,1)$. Based on numerical experiments, monotonic…

Let $N>1$ and $0<\varepsilon\ll 1$. Suppose that the initial vorticity $\omega_0^\varepsilon$ is decomposed into $N$ blobs, each with circulation $\gamma>0$, sharply concentrated a…

Radial-profile conjecture. After a suitable scaling, the limit of the maximizers should be a radially decreasing function with respect to some point.

Uniqueness conjecture for maximizers. At least for some special rearrangement classes, there exists $\hat v\in\mathcal M$ such that

Let $u$ be the function of the vortex separation defined by … where $V$ is the potential governing the approximate Hamiltonian dynamics of two vortices and $Z_1(\tau),Z_2(\tau)$ ar…

Let $omega_0$ be an initial vorticity and let $Omega_+(\omega_0)$ denote its $omega$-limit set. Consider a configuration of $n$ fixed-shape vortex blobs whose centers of mass inter…

A single vortex is the point-vortex dynamical system with one vortex on a compact Riemannian surface. Let the surface have constant curvature and genus greater than one. Koiller's…

Let $N$ coaxial vortex rings have cylindrical-coordinate positions…

Twelve-vortex branch conjecture. This branch is born stable and minimises $H$ for fixed small positive values of $\mu$.

Ten-vortex asymmetric-branch conjecture. The stable branch that minimises $H$ for small fixed values of $\mu$ is asymmetric.

Eight-vortex asymmetric-branch conjecture. An asymmetric relative equilibrium gains stability around these parameter values.

Seven-vortex branch conjecture. This branch minimises $H$ for fixed and small values of the momentum.

Five-vortex stable-branch conjecture. This branch is born stable close to the triangular bipyramid and is the global minimiser of $H$ for fixed values of $\mu>0$ close to zero.

Seven-vortex ground-state conjecture. The pentagonal bipyramid is the minimiser of $H$.

Stuart's wavelength-doubling conjecture. Instability for double-periodic perturbations might hold for the whole family of Kelvin–Stuart vortices, with $\epsilon$ ranging from $0$ t…

Let $\nu$ be the viscosity, $\omega_\alpha$ the vorticity components, and $\mathcal E$ the enstrophy dissipation appearing in … In the turbulent limit, the paper studies dissipatio…