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Existence of global solutions for a class of vector fields on the three-dimensional torus. (English) Zbl 1403.35081
Summary: This work deals with global solvability of a class of vector fields of the form \(\mathsf{L} = \partial / \partial t +(a(x) + i b(x))(\partial / \partial x + \lambda \partial / \partial y)\), where \(a, b \in \mathcal{C}^\infty(\mathbb{T}^1, \mathbb{R})\) and \(\lambda \in \mathbb{R}\), defined on the three-dimensional torus \(\mathbb{T}^3_{(x, y, t)} \simeq \mathbb{R}^3 / 2 \pi \mathbb{Z}^3\). In addition to the interplay between the order of vanishing of the functions \(a\) and \(b\), the change of sign of \(b\) between two consecutive zeros of \(a + i b\) has influence in the global solvability. Also, a Diophantine condition appears in a natural way in our results.

MSC:
35F05 Linear first-order PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B10 Periodic solutions to PDEs
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