On the Hydrodynamic Limit of Ginzburg-Landau Wave Vortices

In this paper, we study the hydrodynamic limit of the finite Ginzburg-Landau wave vortices, which was established in [15]. Unlike the classical vortex method for incompressible Euler equations, we prove here that the densities approximated by the vortex blob method associated with the Ginzburg-Landau wave vortices tend to the solutions of the pressureless compressible Euler-Poisson equations. The convergence of such approximation is proven before the formation of singularities in the limit system as the blob sizes and the grid sizes tend to zero at appropriate rates.