Vortex Solutions of the Ginzburg-Landau Equation in a Thin Domain

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The Ginzburg-Landau equation is a macroscopic model which describes superconducting phenomena. This equation is derived by taking the first variation of the Ginzburg-Landau energy functional and it has the unknown variables of a complex-valued order parameter and a vector potential of magnetic field. We studied the Ginzburg-Landau equation in a 3-dimensional thin domain without an applied magnetic field. We assume that the thickness can be controlled and consider the limiting behavior as the thickness vanishes. The formal reduction tells that in the limit the equation can be reduced to a simpler one without the magnetic effect. We proved by using a perturbation method that if the reduced equation has a non-degenerate stable solution, then the original equation in the thin domain has also stable solution. We also give an explicit example of the domain allowing a non-degenerate stable vortex solution.We also studied the motion law of vortices arising in a gradient system of a simplified Ginzburg-Landau functional in a simply connected 2-dimensional bounded domain. That is a semilinear heat equation of only the order parameter. We derive an explicit form of a singular limit equation as the parameter goes to infinity. By virtue of this explicit form we revealed some dynamical properties of vortices.