an:06943130
Zbl 1403.35081
Bergamasco, Adalberto P.; Dattori da Silva, Paulo L.; Gonzalez, Rafael B.
Existence of global solutions for a class of vector fields on the three-dimensional torus
EN
Bull. Sci. Math. 148, 53-76 (2018).
00414663
2018
j
35F05 35A01 35B10
condition (\(\mathcal{P}\)); Fourier series; Diophantine condition
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.