Stability and motion laws for Ginzburg-Landau vortices on manifolds under dissipative and conservative dynamics

We consider the Ginzburg-Landau energy on compact, simply-connected 2-manifolds. Here we mainly address three problems that involve vortices. One is the stability or instability of critical points of the Ginzburg-Landau energy in the sense of positivity or not of the second variation. A second is the vortex dynamics for the Ginzburg-Landau heat flow, both in the asymptotic regime where the parameter epsilon attends to zero and for fixed epsilon. The third is a similar analysis of vortex motion for the Gross-Pitaevskii equation.;Our first main result is that for compact, simply connected 2-manifolds without boundary, any non-constant critical points must be unstable when epsilon is small if at least one limiting vortex is located at a point of positive Gauss curvature. Furthermore, on a surface of revolution with non-zero Gauss curvature at at least one of the poles, we argue that all critical points are unstable for small epsilon, regardless of the curvature at the limiting vortex locations.;For the Ginzburg-Landau heat flow, we show the vortices of a solution evolve according to the gradient flow of the renormalized energy. We then specialize to the case on a sphere and study the limiting system of ODE's and establish an annihilation result. After that we return to the Ginzburg-Landau heat flow on a sphere and derive some weighted energy identities. Finally we prove the annihilation result for the PDE setting on a surface of revolution with boundary.;For the Gross-Pitaevskii equation, we show the vortices of a solution follow the Hamiltonian point-vortex flow associated with the renormalized energy. Then on surfaces of revolution, we find rotating periodic solutions to the generalized point-vortex problem and seek a rotating solution to the Gross-Pitaevskii equation having vortices that follow those of the point-vortex flow for epsilon sufficiently small.