Robert Jerrard
Professor University of Toronto, Department of Mathematics
University of Toronto, Department of Mathematics
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Articles
On the regularity of timelike extremal surfaces
Communications in Contemporary Mathematics2015
We study a class of timelike weakly extremal surfaces in flat Minkowski space ℝ1+ n, characterized by the fact that they admit a C1 parametrization (in general not an immersion) of a specific form. We prove that if the distinguished parametrization is of class Ck, then the surface is regularly immersed away from a closed singular set of Euclidean Hausdorff dimension at most 1+ 1/k, and that this bound is sharp.
Hydrodynamic limit of the Gross-Pitaevskii equation
Communications in Partial Differential Equations2015
We study dynamics of vortices in solutions of the Gross-Pitaevskii equation i∂ tu= Δ u+ ϵ− 2 u (1−| u| 2) on ℝ2 with nonzero degree at infinity. We prove that vortices move according to the classical Kirchhoff-Onsager ODE for a small but finite coupling parameter ϵ.
Accelerating fronts in semilinear wave equations
Rendiconti del Circolo Matematico di Palermo2015
We study dynamics of interfaces in solutions of the equation ε□u+1εfε(u)=0ε◻u+1εfε(u)=0, for fεfε of the form fε(u)=(u2−1)(2u−εκ)fε(u)=(u2−1)(2u−εκ), for κ∈Rκ∈R, as well as more general, but qualitatively similar, nonlinearities. We prove that for suitable initial data, solutions exhibit interfaces that sweep out timelike hypersurfaces of mean curvature proportional to κκ.
Weighted TV minimization and applications to vortex density models
arXiv2015
Motivated in part by models arising from mathematical descriptions of Bose-Einstein condensation, we consider total variation minimization problems in which the total variation is weighted by a function that may degenerate near the domain boundary, and the fidelity term contains a weight that may be both degenerate and singular. We develop a general theory for a class of such problems, with special attention to the examples arising
from physical models.
On the vortex filament conjecture for Euler flows
arXiv2016
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 binormal curvature flow.
