Global well-posedness of the energy-critical complex Ginzburg-Landau equation in exterior domains

Abstract

In this article, we consider an energy-critical complex Ginzburg-Landau equation in the exterior of a smooth compact strictly convex obstacle. We prove the global well-posedness of energy-critical complex Ginzburg-Landau equation in an exterior domain by the concentration-compactness/rigidity theorem method. As corollaries of our main result, we establish both the existence of global weak solutions to energy-critical defocusing nonlinear Schr\"odinger equations and the global well-posedness theory for energy-critical semi-linear heat equation in exterior domains.