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

Author(s)

Silva, Paulo L. Dattori da ^[1] ; Gonzalez, Rafael B. ^[2] ; Silva, Marcio A. Jorge ^[3]

Affiliation

^[1] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Dept Matemat, Caixa Postal 668, BR-13560970 Sao Carlos, SP - Brazil

^[2] Univ Estadual Maringa, Dept Matemat, BR-87020900 Maringa, PR - Brazil

^[3] Univ Estadual Londrina, Dept Matemat, Caixa Postal 10-011, BR-86057970 Londrina, PR - Brazil

• Document type: Journal article
• Source: Journal of Mathematical Analysis and Applications; v. 492, n. 2 DEC 15 2020.
• Web of Science Citations: 0

Abstract

Let L = partial derivative(t) + a(x)partial derivative(x) be a real vector field defined on the two-dimensional torus T-2, where a is a real-valued and smooth function on T-1. We deal with the global solvability of equations in the form Lu + pu = f, where p, f is an element of C-infinity(T-2). 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 T-2 and, also, in the space of Schwartz distributions D'(T-2). (C) 2020 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 18/14316-3 - Geometric theory of PDE and multidimensional complex analysis
• Grantee: Paulo Domingos Cordaro
• Support Opportunities: Research Projects - Thematic Grants