Solvability for perturbations of a class of real vector fields on the two-torus

Let L=∂t+a(x)∂x be a real vector field defined on the two-dimensional torus T2, where a is a real-valued and smooth function on T1. We deal with the global solvability of equations in the form Lu+pu=f, where p,f∈C∞(T2). Solvability to the equation Lu=f is well-understood. We show that a perturbation...

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Bibliographic Details
Published in:	Journal of mathematical analysis and applications Vol. 492; no. 2; p. 124467
Main Authors:	Dattori da Silva, Paulo L., Gonzalez, Rafael B., Jorge Silva, Marcio A.
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 15-12-2020
Subjects:	Closed range Diophantine conditions Flat functions Fourier series Global solvability Perturbations Perturbations Fourier series Diophantine conditions Closed range Flat functions Global solvability
Online Access:	Get full text
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Summary:	Let L=∂t+a(x)∂x be a real vector field defined on the two-dimensional torus T2, where a is a real-valued and smooth function on T1. We deal with the global solvability of equations in the form Lu+pu=f, where p,f∈C∞(T2). Solvability to the equation Lu=f is well-understood. We show that a perturbation of zero order may affect the global solvability of L; we may maintain, gain or lose solvability by adding a perturbation. This phenomenon is linked to the order of vanishing of the coefficient a of L. We obtained results in the class of smooth functions on T2 and, also, in the space of Schwartz distributions D′(T2).
ISSN:	0022-247X 1096-0813
DOI:	10.1016/j.jmaa.2020.124467