Curve-Supported Concentrations of Vortices and Screw Dislocations Subject to Confinement

We study the concentration of logarithmically interacting defects, such as Ginzburg–Landau vortices and screw dislocations, on closed curves in two dimensions, subject to a confining field. We prove that a measure supported on a smooth closed curve solves the mean-field transport equation exactly when, in Lagrangian labels, the curve moves with the negative induced velocity and each label keeps its mass. This contour law is a gradient flow of the mean-field energy. We prove that the gradient of the energy on a curve has a leading term proportional to the operator $|D|$ with Fourier symbol $|k|$, which prevents every smooth equilibrium curve from being a local minimizer, in a gradient-flow analogue of the Kelvin–Helmholtz instability of vortex sheets. In a radially symmetric case with quadratic confinement, we find an explicit circular equilibrium whose linearization is diagonal in Fourier modes, and we show that, as $N\rightarrow \infty$, the linear spectrum of the regular $N$-gon converges, mode by mode, to that of the circle. Finally, we compute a critical point on two concentric circles with lower energy than the single circle, indicating a rich family of multi-circle equilibria.