Lorentz space estimates and applied boundary current dynamics for Ginzburg-Landau.

摘要

This thesis, which is divided into three parts, is concerned with the study of the Ginzburg-Landau energy functional in two dimensions. In the first part, we prove novel lower bounds for the Ginzburg-Landau energy with or without magnetic field. These bounds rely on an improvement of the "vortex balls construction" estimates by extracting a new positive term in the energy lower bounds. This extra term can be conveniently estimated through a Lorentz space norm, on which it thus provides an upper bound. The Lorentz space we use is critical with respect to the expected vortex profiles and can serve to estimate the total number of vortices and get improved convergence results.;The second component extends the ideas and techniques of the first to the case of the Ginzburg-Landau energy with external magnetic field hex in certain interesting regimes of hex . This allows us to show that for configurations close to minimizers or local minimizers of the energy, the vorticity mass of the configuration (u, A) is comparable to the L2,infinity Lorentz space norm of ▿Au, the covariant derivative. We also establish convergence of the gauge-invariant Jacobians (vorticity measures) in the dual of a function space defined in terms of Lorentz and Lorentz-Zygmund spaces.;For the third component, we turn to an analysis of the time-dependent Ginzburg-Landau equations, subject to an applied electric field on the boundary. The first results show that the equations are well-posed when an applied electric field is present. We continue with an analysis of the evolution of the energy for solutions, deriving various estimates. Finally, we derive the equations governing the dynamics of the system in the limit.