Refined Jacobian Estimates and Gross–Pitaevsky Vortex Dynamics

We study the dynamics of vortices in solutions of the Gross–Pitaevsky equation $${ i u_t = \Delta u + { 1 \over {\varepsilon^2}} u \left(1 - |u|^2 \right)}$$ in a bounded, simply connected domain $${\Omega\subset {\mathbb{R}^2}}$$ with natural boundary conditions on ∂Ω. Previous rigorous results have shown that for sequences of solutions $${u_\varepsilon}$$ with suitable well-prepared initial data, one can determine limiting vortex trajectories, and moreover that these trajectories satisfy the classical ODE for point vortices in an ideal incompressible fluid. We prove that the same motion law holds for a small, but fixed $${\varepsilon}$$ , and we give estimates of the rate of convergence and the time interval for which the result remains valid. The refined Jacobian estimates mentioned in the title relate the Jacobian J(u) of an arbitrary function $${u\in H^1(\Omega;\mathbb{C})}$$ to its Ginzburg–Landau energy. In the analysis of the Gross–Pitaevsky equation, they allow us to use the Jacobian to locate vortices with great precision, and they also provide a sort of dynamic stability of the set of multi-vortex configurations.