Existence of periodic solutions for first-order partial differential equations

• Grant number: 20/14135-9
• Support Opportunities: Scholarships in Brazil - Master
• Start date: July 01, 2021
• End date: February 28, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Adalberto Panobianco Bergamasco
• Grantee: Nguyen Thi Hoang Yen
• Host 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)

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

YEN, Nguyen Thi Hoang. Regularity of periodic solutions for first order partial differential operators. 2023. Master's Dissertation - Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB) São Carlos.