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

TL;DR

This paper proves the global well-posedness of the energy-critical complex Ginzburg-Landau equation in exterior domains, using concentration-compactness methods, and extends results to related nonlinear Schrödinger and heat equations.

Contribution

It establishes the first global well-posedness results for energy-critical complex Ginzburg-Landau equations in exterior domains, employing concentration-compactness and rigidity techniques.

Findings

Proves global well-posedness in exterior domains for the energy-critical complex Ginzburg-Landau equation.

Establishes existence of global weak solutions for energy-critical defocusing nonlinear Schrödinger equations.

Provides global well-posedness results for energy-critical semi-linear heat equations 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.