Global dynamics of energy-critical complex Ginzburg–Landau systems in exterior domains

This paper establishes global well-posedness for the energy-critical complex Ginzburg-Landau (CGL) system in exterior domains with Dirichlet boundary conditions. Working in the complement of a smooth, compact, strictly convex obstacle in $\mathbb{R} ^{N}$ for $N \geq 3$ , we address the fundamental analytical challenges: we develop a concentration-compactness framework for a system that addresses three fundamental challenges: geometric constraints of the boundary, coupled nonlinear interactions, and the lack of translation invariance. By constructing a vector value concentration-compactness framework, we prove the existence of global solutions in the space $\dot{H}^{1}_{D}(\Omega ) \times \dot{H}^{1}_{D}(\Omega )$ with energy decay as $t \to \infty $ . The proof combines several innovative elements: refined Strichartz estimates for the Dirichlet Laplacian, a profile decomposition specifically adapted to exterior domains, and precise analysis of the energy-critical nonlinear coupling mechanisms. Beyond immediate results, our approach yields two significant outcomes for the related system: the existence of global weak solutions for the energy-critical defocusing NLS system and global well-posedness for the energy-critical semilinear heat system in exterior domains. This work constitutes the first comprehensive analysis of energy-critical CGL systems in non-Euclidean geometries, extending previous results for both the scalar CGL equation and the NLS equation in simpler settings. The mathematical techniques developed herein provide a foundation for studying critical nonlinear phenomena in geometrically constrained domains across various fields of analysis.