2026/02/10 by Fernando de Ávila Silva, Marco Cappiello, Alexandre Kirilov +1
Mathematics · Physics and Astronomy · #Nonlinear Differential Equations Analysis #Nonlinear Waves and Solitons #Spectral Theory in Mathematical Physics #math.AP #msc:35B10 #msc:46F05 #msc:58A10 #msc:58J10
paper · pdf · doi:10.1007/s00028-026-01231-9
published as Journal of Evolution Equations, 2026 · 16 pages
arxiv created 2026/02/10 · openalex publication_date 2026/06/23 · openalex created_date 2026/06/24 · openalex updated_date 2026/07/23 · arxiv updated 2026/08/03
Abstract We study the global solvability of a class of differential complexes on the product manifold \mathbb Tm × \mathbb Rn <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> associated with systems of evolution operators of the form Lr = ∂ tr + iar(t)P(x,Dx), r=1,… ,m, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>∂</mml:mi> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> where the coefficients ar <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:math> are real-valued Gevrey functions on the torus and P(x,Dx) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is a globally elliptic normal differential operator on \mathbb Rn <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> . Within the framework of time-periodic Gelfand-Shilov spaces, we introduce a natural differential complex generated by these operators and investigate its solvability in both functional and ultradistributional settings. We provide a complete characterization of global solvability in terms of a Diophantine condition involving the constant part of the associated 1-form and the spectrum of P . We also analyze global hypoellipticity of the complex. These results extend previous works on scalar operators and constant coefficient systems to the setting of differential complexes with time-dependent real coefficients.