Stability in $2D$ Ginzburg-Landau passes to the limit Sylvia Serfaty 35B3835B4035B3535B2535Q99Ginzburg-Landauvorticesstabilityasymptotics We prove that if we consider a family of \textit{stable} solutions to the Ginzburg-Landau equation, then their vortices converge to a \textit{stable} critical point of the ``renormalized energy.'' Moreover, in the case of instability, the number of ``directions of descent'' is bounded below by the number of directions of descent for the renormalized energy. A consequence is a result of nonexistence of stable nonconstant solutions to Ginzburg-Landau with homogeneous Neumann boundary condition. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2497 10.1512/iumj.2005.54.2497 en Indiana Univ. Math. J. 54 (2005) 199 - 222 state-of-the-art mathematics http://iumj.org/access/