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Existence of periodic solutions for first-order partial differential equations

Grant number: 20/14135-9
Support type:Scholarships in Brazil - Master
Effective date (Start): July 01, 2021
Effective date (End): June 30, 2023
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Analysis
Principal researcher:Adalberto Panobianco Bergamasco
Grantee:Nguyen Thi Hoang Yen
Home Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
Associated research grant:18/14316-3 - Geometric theory of PDE and multidimensional complex analysis, AP.TEM

Abstract

The project's theme is the study of necessary and sufficient conditions for the existence of global solutions to a first order linear partial differential equation.We study equations of the form Lu=f , where L is a complex vector field on the n-dimensional torus. Let X denote the space of complex-valued functions defined on the n-dimensional torus.We work in the context of smooth functions and vector fields; here, smooth stands for infinitely differentiable.The problem under study is to find out if, for every function f in X, verifying natural conditions, there exists a function f in X such that Lu=f. When the answer is in the affirmative, one says that L is globally solvable.Equivalently, one wants to know if the operator L, acting from X to X, has closed range.An article, published in the Journal of Fourier Analysis and Applications in 2017, studied the problem of global solvability for a class of complex vector fields. The authors of this article are A. Bergamasco, P. Dattori da silva and R. Gonzales.In this class of vector fields, one of the variables plays a special role; in particular, the coefficient of the vector fields depend only upon such a variable. We are in the so-called tube case.The main theorem links global solvability to concepts coming from several areas of Mathematics.One of the main goals of this project is that the student be able to understand and provide details of the proofs in the above-mentioned article.As a second step, one intends to analyze the effect of lower-order perturbations, such as by pseudodifferential operators, on the validity of global solvability.An alternative possibility for an extension of the work is the study of solvability in other spaces of functions and distributions. (AU)

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