Inserted: 7 sep 2010
Last Updated: 6 sep 2012
In this paper we analyze the hydrodynamic equations for Ginzburg-Landau vortices as derived by Weinan E. In particular, we are interested in the mean-field model describing the evolution of two patches of vortices with equal and opposite degrees. Many results are already available for the case of a single density of vortices with uniform degree. This model does not take into account the vortex annihilation, hence it can also be seen as a particular instance of the Chapman-Rubinstein-Schatzman formulation. We establish global existence of $L^p$ solutions, exploiting some optimal transport techniques. We prove uniqueness for $L^\infty$ solutions, as expected by analogy with the incompressible Euler equations in fluidodynamics. We also consider the corresponding Dirichlet problem in a bounded domain. Moreover, we show some simple examples of $1$-dimensional dynamic.