Viscosity solutions of a degenerate parabolic-elliptic system arising in the mean-field theory of superconductivity
The authors consider the mathematical model of the mean-field theory of superconducting vortices described by the degenerate parabolic-elliptic system \[ \Psi_t-| \nabla\Psi|\Biggl( \sigma\nabla \Biggl({\nabla\Psi\over |\nabla\Psi|}+ q- \Psi\Biggr)\Biggr)= 0\quad\text{in }\Omega\times ]0,T[, \] \[ -\Delta q+ \kappa_\Omega q= \kappa_\Omega\Psi\quad\text{in }\mathbb{R}^2\times ]0,T[, \] together with a homogeneous Neumann condition for \(\Psi\) on \(\partial\Omega\times ]0,T[\) and Neumann condition \(q_\infty\) for \(q\) at infinity. Here \(\Omega\) is a bounded domain in \(\mathbb{R}^2\) and \(\sigma\) is a nonnegative constant. The existence and uniqueness of the solution are proved. The dynamic equation is solved in the viscosity sense while the elliptic equation is solved in the weak sense. The comparison principle is proved for the dynamic system. For \(\sigma= 0\), it is proved that \((\omega, H)= (\nabla^\perp\Psi, \nabla^\perp q)\) is the local weak solution of the model of the mean-field theory of superconductivity. (Here \(H\) is a magnetic field and \(\omega\) is the averaged vorticity.) In addition, the stationary problem is investigated for \(\sigma= 0\) and domains \(\Omega\) constructed that allow for solutions with more than two superconducting phases.
mean-field theory of superconducting vortices
scientific article; zbMATH DE number 1287914