  We establish the existence of a global solution for a new family of
fluid-like equations, which are obtained in a joint work with Serfaty in
certain regimes as the mean-field evolution of the supercurrent density in a
(2D section of a) type-II superconductor with pinning and with imposed electric
current. We also consider general vortex-sheet initial data, and investigate
the uniqueness and regularity properties of the solution. For some choice of
parameters, the equation under investigation coincides with the so-called lake
equation from 2D shallow water fluid dynamics, and our analysis then leads to a
new existence result for rough initial data.
