Stability of solutions for 2D anisotropic gradient flows and Blow-Up Conditions in Some 2D Fluid Equations

• Grant number: 25/08196-9
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: October 01, 2025
• End date: September 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Lucas Catão de Freitas Ferreira
• Grantee: Ricardo Martins Mendes Guimarães
• Supervisor: Jose Antonio Carrillo de La Plata
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: University of Oxford, England
• Associated to the scholarship: 24/00744-4 - Study of possible scenarios of singularity formation for generalized surface quasi-geostrophic equations, BP.PD

Abstract

Our primary goal in this project is to investigate the stability of global minimizers among solutions of gradient flow equations for a broad class of two-dimensional anisotropic interaction energies, in which Riesz¿type singular repulsive kernels are coupled with a smooth confining potential and a directional weight that governs anisotropy. We aim to describe under which conditions, depending on the precise parameter intervals, that give rise to distinct potential minimizer behaviors and their stability. Our second goal is to study possible scenarios of singularity formation for the MHD equations in both two and three dimensions, as well as for generalized dissipative surface quasi-geostrophic equations with a velocity field that is logarithmically more singular than the velocity field of the standard generalized SQG equation. For these equations, questions regarding global well-posedness and finite-time blow-ups remain open. In this context, our objective is to establish results concerning the occurrence or exclusion of finite-time singularities, depending on the geometry of the level set structures of the solution. (AU)