×

zbMATH — the first resource for mathematics

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.
MSC:
35F05 Linear first-order PDEs
35N10 Overdetermined systems of PDEs with variable coefficients
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bergamasco, A. P.; Dattori, P. L.; Gonzales, R. B., Global solvability and global hypoellipticity in Gevrey classes for vector fields on the torus, J. Differ. Equ., 264, 5, 3500-3526 (2018) · Zbl 1395.35074
[2] Bergamasco, A. P.; Dattori da Silva, P. L.; Gonzales, R. B., Existence and regularity of periodic solutions to certain first-order partial differential equations, J. Fourier Anal. Appl., 23, 65-90 (2017) · Zbl 1357.35020
[3] Bergamasco, A. P.; de Medeira, C.; Kirilov, A.; Zani, S. L., On the global solvability of involutive systems, J. Math. Anal. Appl., 444, 1, 527-549 (2016) · Zbl 1359.37107
[4] Bergamasco, A. P.; de Medeira, C.; Zani, S. L., Globally solvable systems of complex vector fields, J. Differ. Equ., 252, 4598-4623 (2012) · Zbl 1242.35092
[5] Bergamasco, A. P.; Petronilho, G., Global solvability of a class of involutive systems, J. Math. Anal. Appl., 233, 1, 314-327 (1999) · Zbl 0942.35011
[6] Cardin, F.; Gramchev, T.; Lovison, A., Exponential estimates for oscillatory integrals with degenerate phase functions, Nonlinearity, 21, 3, 409-433 (2008) · Zbl 1132.41342
[7] Dattori da Silva, P. L.; Meziani, A. A., Gevrey differential complex on the torus, J. Fourier Anal. Appl., 26, 8, 712-732 (2020) · Zbl 1434.58008
[8] Droste, B., Beitrag zum divisionsproblem for ultradistributionen und ein fortsetzungssatz, Manuscr. Math., 27, 259-278 (1979) · Zbl 0401.46020
[9] Greenfield, S. J.; Wallach, N. R., Global hypoellipticity and Liouville numbers, Proc. Am. Math. Soc., 31, 112-114 (1972) · Zbl 0229.35023
[10] Hannah, H.; Himonas, A. A.; Petronilho, G., Gevrey regularity of the periodic gkdv equation, J. Differ. Equ., 250, 5, 259-278 (2011) · Zbl 1209.35117
[11] Hounie, J.; Zugliani, G., Global solvability of real analytic involutive systems on compact manifolds, Math. Ann., 369, 1177-1209 (2017) · Zbl 1380.35129
[12] Junior, A. A.; de Medeira, C.; Kirilov, A., Global Gevrey hypoellipticity on the torus for a class of systems of complex vector fields, J. Math. Anal. Appl., 474, 1, 712-732 (2019) · Zbl 1418.35087
[13] Petronilho, G., Global s-solvability, global s-hypoellipticity and Diophantine phenomena, Indag. Math., 16, 6, 67-90 (2005) · Zbl 1065.35100
[14] Petronilho, G.; Zani, S. L., Global s-solvability and global s-hypoellipticity for certain perturbations of zero order of systems of constant real vector fields, J. Differ. Equ., 244, 9, 2372-2403 (2008) · Zbl 1155.35010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.