Lagrangian solutions for the transport and continuity equations

• Grant number: 24/21041-1
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: May 01, 2025
• End date: April 30, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Hermano Frid Neto
• Grantee: Henrique Borrin de Souza
• Host Institution: Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP). Universidade de São Paulo (USP). Ribeirão Preto , SP, Brazil

Abstract

The study of Lagrangian solutions, that is, functions which can be written as the transport of the initial data through a flow, is becoming a feasible alternative for obtaining renormalized solutions. In this project, we expose our goals of extending the state of the art due to Nguyen (ARMA, 2021) in order to obtain a regular renormalized flow for vector fields which can be written as a convolution of truncated singular kernel and a bounded variation density; the candidate was able to prove in the Sobolev density case (JDE, 2025). Moreover, our goal is to obtain a local version of Lagrangian solutions for the damped continuity equation of Colombo-Crippa-Spirito (CVPDE, 2015) without the growth assumption seminally introduced by DiPerna-Lions (Invent. Math., 1989). Finally, if it of interest for extending the fellowship for one more year, we would like to extend the ideas of Fernández-Real (Comm. Math. Phys., 2018) for the Vlasov-Biot-Savart system (previously studied by the candidate and Marcon (M2AS, 2022) in tridimensional physical domain) for bounded domains with (physically) idealized boundary. (AU)