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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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##### References:
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