<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">R</mi> <mi>N</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{R} ^{N}$</EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </math></EquationSource> <EquationSource Format="TEX">$N \geq 3$</EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <msubsup> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> <mi>D</mi> <mn>1</mn> </msubsup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msubsup> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> <mi>D</mi> <mn>1</mn> </msubsup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\dot{H}^{1}_{D}(\Omega ) \times \dot{H}^{1}_{D}(\Omega )$</EquationSource> </InlineEquation> with energy decay as <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </math></EquationSource> <EquationSource Format="TEX">$t \to \infty $</EquationSource> </InlineEquation>. 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.</p>

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

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2025年11月21日

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Kalim Ullah,
Muhammad Ishfaq Khan,
Sabiha Zehra,
Homan Emadifar,
Karim K. Ahmed

摘要

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 $R^{N}$ $\mathbb{R} ^{N}$ for $N \geq 3$ $N \geq 3$ , 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 ${\dot{H}}_{D}^{1} (Ω) \times {\dot{H}}_{D}^{1} (Ω)$ $\dot{H}^{1}_{D}(\Omega ) \times \dot{H}^{1}_{D}(\Omega )$ with energy decay as $t \to \infty$ $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.