Vorticity measure of type-II superconducting thin films (Q2251736)

The paper studies the asymptotic behavior of vorticity measures associated to the vortices of the solution that describes the distribution of vortices in a simply connected domain. First, for a considered thin structure placed in an applied magnetic field, the full 3D Ginzburg-Landau model is reduced to a simplified model that retains the basic features of the vortex state referred to as (G.L.). The solutions of the obtained system (G.L.) are the critical points of the Ginzburg-Landau energy \(J(u)\), where \(u\) is the complex superconducting order parameter which indicates the local state of the material. The vortex presents itself an isolated zero of the order parameter \(u\), around which \(u\) has a non-zero winding number. Then, a parameter \(\varepsilon\) is defined through the material parameter \(\kappa\) that determines the type of superconducting material as \(\varepsilon=1/\kappa\). The paper addresses the question of the general behavior of critical points, i.e., the asymptotic behavior, as \(\varepsilon\to0\), of solutions of the system (G.L.) that are not necessary global or local minimizers. The paper considers intermediate fields of the order \(|\ln\varepsilon|\). It is assumed that the energy blows up like the square of the applied magnetic field. This assumption is automatically satisfied for the minimum of the energy. The proofs use, with some modifications, the technique of the paper by \textit{E. Sandier} and \textit{S. Serfaty} [Duke Math. J. 117, No. 3, 403--446 (2003; Zbl 1035.82045)]. Moreover, the fact that the vector potential \(A\) is given means that the field appearing in Ampere's law is not the curl of \(A\). This leads to a change in the definition of the appropriate stress tensor. For this tensor a convergence in finite part analysis is done, with a limit that is not divergence free. So, first, the convergence of the induced magnetic field is derived except on a set of small perimeters. Then, the asymptotic behavior of the stress-energy tensor, associated to the solution \(u\) of the (G.L.), is studied. Finally, the authors investigate the property of the limiting vorticity-measure.