We study the gradient flow equations derived from an Einstein-Maxwell-Higgs model in 3+1 dimensions. We see how this model relates to a phenomenological description of a superconductor in two ways. In flat spacetime the model is equivalent to the Ginzburg-Landau theory of superconductivity and describes a 3 dimensional superconductor. In curved spacetime with negative cosmological constant, we can apply the AdS/CFT correspondence to obtain a 2 dimensional theory on the boundary that describes a superconductor. The gradient flow equations in both cases are a system of parabolic partial differential equations analagous to the heat equation. The flow describes a non-isolated system where energy is allowed to dissipate as the system evolves towards thermal equilibrium. In the first case the gradient flow gives rise to the time-dependent Ginzburg-Landau equations, and we study the formation and interaction of superconducting vortices. In the second case, the flow in the bulk describes the formation of scalar hair around a black hole, which corresponds to the formation of a superconducting condensate on the boundary. The flow in the bulk creates an equivalent flow on the boundary that can be thought of as an extension of the AdS/CFT correspondence to non-equilibrium configurations.