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Global Gevrey solvability for a class of perturbations of involutive systems. (English) Zbl 1454.35050
In this article, the author is interested in the global solvability, in the Gevrey meaning, on the \(n\)-dimensional torus \(\mathbb T^n\) of the system \[(L_j=\partial_{t_j}+a_j(t)\partial_x+b_j(t))_{1\leq j\leq n}\in\mathbb T_t^n\times S_x^1,\] where \(a_j\in G^s(\mathbb T^n,\mathbb R)\) and \(b_j\in G^s(\mathbb T^n)\) are both \(s\)-Gevrey on the torus \(\mathbb T^n\), and where \(\displaystyle\sum_{j=1}^n a_jdt_j\) and \(\displaystyle\sum_{j=1}^n b_jdt_j\) are both closed. More precisely, he focuses in the following question: supposing that the system \[(\partial_{t_j}+a_j(t)\partial_x)_{1\leq j\leq n}\] is globally \(s\)-solvable, when the system \((L_j)_{1\leq j\leq n}\) is also globally \(s\)-solvable?
To do that, the author first reduces the study of the system \((L_j)_{1\leq j\leq n}\) to the study of a system whose the principal part has constant coefficients. Then, he generalizes the results of [G. Petronilho and S. L. Zani, J. Differ. Equations 244, No. 9, 2372–2403 (2008; Zbl 1155.35010)] in order to characterize the global \(s\)-solvability of the latter system.
35F05 Linear first-order PDEs
35N10 Overdetermined systems of PDEs with variable coefficients
Full Text: DOI
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