Existence of Weak Solutions to Some Vortex Density Models

Abstract

We study the weak solutions to equations arising in the modeling of vortex motions in superfluids such as superconductors. The global existence of measure-valued solutions is established with a bounded Radon measure as initial data. Moreover, we get a local space-time $L^q$ estimate for the continuous part of the solution, and we prove the global existence of a distributional weak solution for a particular case. We also consider a modification to the model in order to physically account for the different signs of vortices, and we present, in one space dimension, the global existence of weak solutions with the initial data in BV for the modified model.